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Qm Three Dimensional Scattering

1 · Prerequisites

2 · Summary

This page develops the pair's three-dimensional scattering model for the spin-zero massive one-particle sector. The model is an explicitly adopted assumption of the pair, The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units: the state space is $H_{0} = L^{2} (R^{3}; C)$ with the position and momentum families of the framework, the potential $V$ is prescribed, not derived, and two declared short-range classes are fixed — the bounded compact-support class and the resolvent-tail class $V = V_{2} + V_{b}$ with $V_{2} \in L^{2}$ , $V_{b}$ bounded and the shifted nondimensional free resolvent tail integrable. The free and interacting assigned energy observables $H^{f r e e}$ and $H = H^{f r e e} + M_{V}$ are read on the common Sobolev domain $W^{2, 2} (R^{3}; C)$ , energy in joules, and the wave-operator, scattering-operator and asymptotic-completeness vocabulary is declared in the same item. The pure Coulomb potential is explicitly outside both classes and is treated by the separate long-range model of this page.

The local mathematics precedes the physical readings. Unit, time and spectral correspondence between the physical scattering operators and the nondimensional Laplacian supplies positive energy scaling. The model also fixes a reference length $L_{*}$ , dimensionless $y = x / L_{*}$ , $E_{*} = ℏ^{2} / (2 m L_{*}^{2})$ and the unitary $(D_{*} f) (y) = L_{*}^{3 / 2} f (L_{*} y)$ . In the tail criterion the dimensionless pair is $A_{0, *} = - Δ_{y}^{w e a k}$ , $A_{*} = A_{0, *} + M_{W_{*}}$ with $W_{*} (y) = V (L_{*} y) / E_{*}$ ; physical radius $L_{*} r$ corresponds to dimensionless cutoff $r$ . The physical pair is unitarily equivalent to $E_{*} A_{0, *}, E_{*} A_{*}$ and its comparison limits use $τ = t E_{*} / ℏ$ . Stationary formulas instead retain physical $x, k, r$ and $W_{x} = 2 m V / ℏ^{2}$ , in inverse-square-length units; energy division alone does not make these coordinates dimensionless. The three-dimensional continuity identity, the vanishing stationary divergence and the incident and outgoing current readings proves the continuity identity $\partial_{t} n + div j = 0$ for the declared differentiable solutions, the vanishing stationary divergence and sphere flux, and the two explicit current readings: the plane wave carries $j = (ℏ k / m) d$ and the outgoing field $(a (ω) + i b (ω)) e^{i k r} / r$ carries radial density $(ℏ k / m) ∣ f (ω) ∣^{2} / r^{2}$ ; the general outgoing field of the second reading is only $C^{1}$ and is not claimed to solve the free equation.

The stationary elastic beam model The declared stationary elastic beam model: incident plane wave, scattered wave, current readings and differential and total cross sections declares the incident plane-wave reading at wave number $k$ and direction $d$ , the declared number density $n_{i n c} > 0$ and physical incident flux density $J_{i n c} = n_{i n c} ℏ k / m$ , the outgoing asymptotic reading with amplitude $f_{k}$ , the declared generalized stationary solution and the differential and total cross-section readings $d σ / d Ω = ∣ f_{k} ∣^{2}$ and $σ = \int ∣ f_{k} ∣^{2}$ as ratios of current densities under explicitly stated equal-channel flux assumptions. A generalized stationary solution is not an element of the state space and no preparation, beam apparatus or counter appears in the model.

On the dynamical side, Existence of the model's wave operators, isometry and intertwining, and the purely absolutely continuous free spectrum proves, for the compact-support class, the Cook integrability of the potential along the free flow, the existence of the model's wave operators as strong limits of unitaries together with isometry and intertwining, the purely absolutely continuous free spectrum with its spectral density, and the absence of positive eigenvalues of $H$ . Asymptotic completeness and the unitary scattering operator for real bounded compactly supported potentials proves asymptotic completeness for that class: the ranges coincide with $H_{a c} (H) = H_{c} (H)$ , the singular continuous subspace is ${0}$ , and $S = Ω_{+}^{*} Ω_{-}$ is unitary and commutes with the free flow. Asymptotic completeness for the resolvent-tail short-range class, with the decaying-tail example and the explicit exclusion of the Coulomb tail extends the same conclusions to the resolvent-tail class, checks the sufficient decaying-tail form $∣ V_{t a i l} ∣ \leq C (1 + ∣ x ∣ / L_{*})^{- 1 - δ}$ , $δ > 0$ , and records the explicit failure of that sufficient decay for the Coulomb tail.

On the stationary side, The model's stationary scattering wave, its outgoing amplitude and the differential and total cross-section readings constructs the outgoing stationary solution of the beam model, proves the bounded invertibility that defines it, its $H_{l o c}^{2}$ regularity, the uniform far-field expansion with smooth amplitude and $O (r^{- 2})$ remainder and the resulting cross-section readings; The controlled Born reading: operator-norm condition, uniform amplitude error and coupling scaling gives the operator-norm controlled Born approximation with its explicit condition $q = ∥ K_{k} ∥ < 1$ and the coupling-scaling form; The model's partial-wave and phase-shift vocabulary for a declared central potential fixes the channel vocabulary for a declared central potential — the regular and outgoing radial solutions, the unit-modulus channel factor $S_{l} = \overline{W (φ_{l}, f_{l})} / W (φ_{l}, f_{l}) = e^{2 i δ_{l}}$ with $δ_{l}$ read modulo $π$ , and the channel cross-section reading — and Partial-wave expansion, unit-modulus channel factors and the phase-shift cross-section series proves the Legendre expansion, the phase-shift series for the total cross section and the identification of the channel objects with those of the cited mathematics. Identification of the model's scattering operator with the on-shell stationary operator, and the central per-channel form identifies the stationary construction with the dynamical scattering operator: $F_{0} S F_{0}^{*} = M_{S (E)}$ with $S (E) = I + i k A_{k} / (2 π)$ unitary on the energy-angular fiber, the rotational covariance of the on-shell operator and its diagonal channel decomposition for central potentials. The declared single-channel resonance model: pole-shaped channel factor and its cross-section reading and The resonance model's phase-shift law, unitarity and channel cross section declare and analyse a single-channel resonance factor, its induced phase shift, the Lorentzian $\sin^{2} δ_{l}$ with half-height width $Γ$ , and the channel cross-section reading, with no lifetime, decay law or measured line shape asserted.

The long-range case is separate and explicitly modified: The declared long-range Coulomb model and its Dollard-modified free comparison declares the one-centre Coulomb model with $γ = 2 m κ / ℏ^{2}$ and the Dollard-modified comparison $U_{D} (t) = e^{- i t H^{f r e e} / ℏ} h_{t} (H^{f r e e})$ , and Completeness of the Dollard-modified wave operators for the declared Coulomb model proves existence, isometry, intertwining and asymptotic completeness of the modified wave operators and unitarity of $S_{D}$ ; neither existence nor completeness of the ordinary short-range comparison is asserted for the Coulomb potential. Scope and empirical status of the three-dimensional scattering page and Long-range, threshold and model-scope limits of the scattering page collect the scope and empirical status of the page: every statement is a conditional model deduction, the page reports no experiment, no apparatus, no count and no uncertainty budget, and a comparison of these readings with data would require a declared preparation, target, fluence, detector acceptance and background model together with a statistical model for the counts.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Unit, time and spectral correspondence between the physical scattering operators and the nondimensional Laplacian

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $m, ℏ > 0$ , put $c : = ℏ^{2} / (2 m)$ , let $F_{2}$ be the unitary Plancherel transform of Real multiplication operators and free plus bounded-potential Schrödinger operators clause (b) in dimension $d = 3$ , let $V : R^{3} \to R$ be measurable and put $W : = V / c$ . For a measurable $U$ let $M_{U}$ be the maximal multiplication operator on $L^{2} (R^{3}; C)$ , with $D (M_{U}) : = {f \in L^{2} : U f \in L^{2}}$ and $M_{U} f : = U f$ . Write $A_{0} : = F_{2}^{*} M_{4 π^{2} ∣ ξ ∣^{2}} F_{2}, D (A_{0}) : = {f \in L^{2} (R^{3}; C) : ∣ ξ ∣^{2} F_{2} f \in L^{2} (R^{3}; C)},$ $A : = A_{0} + M_{W}, D (A) : = D (A_{0}), H_{0} : = c A_{0}, H : = c A = c A_{0} + M_{V} .$ $D (A_{0}) \subseteq D (M_{V}) = D (M_{W})$ is assumed, and assume that $A = A_{0} + M_{W}$ is self-adjoint on $D (A_{0})$ . These two hypotheses hold automatically for real bounded measurable $V$ by Real multiplication operators and free plus bounded-potential Schrödinger operators clauses (a),(c), so every original bounded-potential case is retained. So $A_{0}$ is the operator $- Δ_{w}$ of The three-dimensional free Schrödinger operator as the weak Laplacian on its Sobolev domain and $D (A_{0}) = W^{2, 2} (R^{3}; C)$ (Integer-order Sobolev spaces and their norms). For a self-adjoint operator $T$ write $E_{T}$ for the regular projection valued measure of Spectral theorem for unbounded self-adjoint operators (PVM form) and $μ_{ψ}^{T} (B) : = ⟨ E_{T} (B) ψ, ψ ⟩$ for $ψ \in L^{2}$ . Then:

(a) $A$ and $H$ are self-adjoint with domain $D (A) = W^{2, 2} (R^{3}; C)$ , and $H = c A$ and $H_{0} = c A_{0}$ hold as operator identities on that common domain;

(b) for every Borel set $B \subseteq R$ , $E_{H} (c B) = E_{A} (B), E_{H_{0}} (c B) = E_{A_{0}} (B);$ consequently $μ_{ψ}^{H} (c B) = μ_{ψ}^{A} (B)$ for every $ψ$ and every Borel $B$ , and $σ (H) = c σ (A)$ , $σ (H_{0}) = c σ (A_{0})$ ;

(c) for every $t \in R$ , with $τ : = ℏ t / (2 m) = \frac{t c}{ℏ}$ , $e^{- i t H / ℏ} = e^{- i τ A}, e^{- i t H_{0} / ℏ} = e^{- i τ A_{0}},$ where the exponentials are those of A self-adjoint operator generates a strongly continuous unitary group;

(d) let $F : R \to B (L^{2})$ be any family of bounded operators and $τ (t) : = ℏ t / (2 m)$ . For every $ψ \in L^{2}$ the strong limit $\lim_{t \to \pm \infty} F (τ (t)) ψ$ exists if and only if $\lim_{τ \to \pm \infty} F (τ) ψ$ exists, and then the two limits are equal; in particular ${s-lim}_{t \to \pm \infty} e^{i t H / ℏ} e^{- i t H_{0} / ℏ} = {s-lim}_{τ \to \pm \infty} e^{i τ A} e^{- i τ A_{0}}$ in the sense that either strong limit exists exactly when the other does, and then they are equal;

(e) $μ_{ψ}^{H}$ is absolutely continuous with respect to Lebesgue measure if and only if $μ_{ψ}^{A}$ is, and likewise for singularity with respect to Lebesgue measure and for absence of atoms; and $ψ \neq 0$ is an eigenvector of $H$ with eigenvalue $λ$ if and only if it is an eigenvector of $A$ with eigenvalue $λ / c$ .

The statement is an operator identity and a statement about spectral measures; no physical interpretation, no scattering limit and no cross section is asserted.

Facts & Assumptions

Given: the parameters $m, ℏ > 0$ and $c = ℏ^{2} / (2 m)$ ; the real measurable $V$ and $W = V / c$ , with $D (A_{0}) \subseteq D (M_{V})$ and $A = A_{0} + M_{W}$ self-adjoint on $D (A_{0})$ ; the operators $A_{0}, A, H_{0}, H$ of the statement; the choice hypotheses.

[F1]

The free operator and its Sobolev domain. With $F_{2}$ as above and $c_{0} : = 2 π^{2} ℏ^{2} / m$ , the operator $F_{2}^{*} M_{c_{0} ∣ ξ ∣^{2}} F_{2}$ on ${f : ∣ ξ ∣^{2} F_{2} f \in L^{2}}$ is self-adjoint, its domain is $W^{2, 2} (R^{3}; C)$ , and it acts as $f \mapsto - \frac{ℏ^{2}}{2 m} Δ_{w} f$ (The three-dimensional free Schrödinger operator as the weak Laplacian on its Sobolev domain clause (a), Integer-order Sobolev spaces and their norms).

[F2]

Free plus bounded potential. For $d \geq 1$ , $c_{0} > 0$ , $b \in R$ and real measurable bounded $V_{0}$ , the operator $F_{2}^{*} M_{c_{0} ∣ ξ ∣^{2} + b} F_{2}$ is self-adjoint on its stated domain, and $H : = H_{0} + M_{V_{0}}$ is self-adjoint on $D (H_{0})$ (Real multiplication operators and free plus bounded-potential Schrödinger operators clauses (b), (c)).

[F3]

Positive scaling of self-adjoint operators. For $c_{0} > 0$ and a densely defined self-adjoint $A$ on a nonzero complex Hilbert space, the operator $c_{0} A$ with domain $D (A)$ is self-adjoint, $ρ (c_{0} A) = c_{0} ρ (A)$ and $σ (c_{0} A) = c_{0} σ (A)$ (Positive scalar multiples of self-adjoint operators: domains, resolvents, spectra and eigenbases clauses (a), (b)).

[F4]

The spectral theorem. Every self-adjoint operator $T$ on a nonzero complex Hilbert space has a unique regular projection valued measure $E_{T}$ on the Borel $σ$ -algebra of $R$ with $D (T) = {x : \int λ^{2} d E_{T, x} (λ) < \infty}$ and $T x = \int λ d E_{T} (λ) x$ for $x \in D (T)$ ; conversely, for a regular projection valued measure $F$ the operator $\int λ d F$ with that domain is self-adjoint with projection valued measure $F$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure).

[F5]

Bounded and unbounded projection valued integrals. For a bounded measurable $g$ the operator $Φ_{E} (g)$ satisfies $⟨ Φ_{E} (g) x, y ⟩ = \int g d E_{x, y}$ and $∥ Φ_{E} (g) ∥ \leq ∥ g ∥_{\infty}$ (Bounded borel pvm integral clauses 1, 2); for measurable $f$ the operator $f (E)$ has domain $D (f (E)) = {x : \int ∣ f ∣^{2} d E_{x} < \infty}$ and for each $x \in D (f (E))$ the vector $f (E) x$ is the Hilbert-space norm limit of the bounded truncation integrals applied to $x$ , with truncations $f 1_{{∣ f ∣ \leq n}}$ (Integral of a measurable function against a projection-valued measure). The same bounded integral also satisfies $∥ Φ_{E} (g) x ∥^{2} = \int ∣ g ∣^{2} d E_{x}$ ; bounded measurable $g_{n}$ with $∣ g_{n} ∣ \leq C ∣ f ∣$ and $g_{n} \to f$ pointwise satisfy $g_{n} (E) x \to f (E) x$ for $x \in D (f (E))$ (The unbounded PVM integral is densely defined, closed and normal, final clause).

[F6]

The exponential as a projection valued integral. For self-adjoint $T$ and $t \in R$ , $e^{i t T} = \int e^{i t λ} d E_{T} (λ)$ is a strongly continuous one-parameter unitary group with generator $i T$ (A self-adjoint operator generates a strongly continuous unitary group); this is the bounded functional calculus of Unbounded Borel functional calculus: domains, products, spectral mapping applied to the bounded continuous functions $λ \mapsto e^{i t λ}$ .

[F7]

Scalar convergence and Lebesgue scaling. If $0 \leq f_{n} ↑ f$ pointwise for measurable functions, $\int f_{n} d μ ↑ \int f d μ$ (Monotone convergence for the integral). For $c > 0$ on $R$ , dilation preserves Lebesgue measurability and $λ (c B) = c λ (B)$ , hence it and its inverse preserve null sets (For a nonzero real $c$ , dilation by $c$ multiplies Lebesgue outer measure by $∣ c ∣^{n}$ , and reflection in the origin preserves it clauses (2),(3), with $n = 1$ ).

Proof

technique · direct; identify the two operators on their common domain, push the projection valued measure forward along $\lambda\mapsto c\lambda$ and use the uniqueness clause of the spectral theorem, then transfer the exponential and the strong limits by the change of variables $\tau=\hbar t/(2m)$

Given: the data of the statement and [F1]-[F7].

1.1F1algebra

Applying [F1] with $m = 1 / 2$ , $ℏ = 1$ gives $c_{0} = 4 π^{2}$ and shows that $A_{0} = F_{2}^{*} M_{4 π^{2} ∣ ξ ∣^{2}} F_{2}$ is self-adjoint with $D (A_{0}) = W^{2, 2} (R^{3}; C)$ and $A_{0} f = - Δ_{w} f$ .

1.2F1F2F3givenalgebra

Multiplying by $c > 0$ , [F3] clause (a) makes $c A_{0} = H_{0}$ self-adjoint on $D (A_{0})$ , and [F1] identifies its action with $- ℏ^{2} Δ_{w} / (2 m)$ . Since $W = V / c$ and $c \neq 0$ , the maximal multiplier domains satisfy $D (M_{W}) = D (M_{V})$ ; the assumed inclusion $D (A_{0}) \subseteq D (M_{V})$ makes both sums well defined on the common domain. The operator $A = A_{0} + M_{W}$ is self-adjoint there by hypothesis, and [F3] makes $H = c A$ self-adjoint on that same domain. For $f \in D (A_{0})$ , $c A f = c A_{0} f + c W f = c A_{0} f + V f$ , proving (a). If $V$ is bounded, $M_{V}$ is bounded on all $L^{2}$ and [F2] proves the assumed self-adjointness of $A$ automatically; this retains the original branch.

1.3F4algebra

Put $m_{c} (λ) = c λ$ and, for Borel $B$ , $F (B) : = E_{A} (m_{c}^{- 1} (B)) = E_{A} (B / c)$ . Preimages preserve intersections and countable disjoint unions, so [F4] gives the projection, identity and strong countable-additivity axioms for $F$ . For every $x$ , $F_{x} (B) = E_{A, x} (B / c)$ . This scalar measure is regular: a compact $K \subseteq B / c$ approximating $E_{A, x} (B / c)$ transports to the compact $c K \subseteq B$ with the same measure, and an open $U \supseteq B / c$ with arbitrarily small excess transports to the open $c U \supseteq B$ with the same excess. These follow from regularity of $E_{A, x}$ and the homeomorphism $m_{c}$ . Thus $F$ is a regular PVM, with the preimage direction appropriate to scaling $A$ by $c$ .

1.4givenalgebra

Let $ψ, y \in L^{2}$ . If $\lim_{τ \to + \infty} F (τ) ψ = y$ , then for $ϵ > 0$ there is $S$ with $∥ F (τ) ψ - y ∥ < ϵ$ for $τ > S$ ; since $τ (t) = ℏ t / (2 m) \to + \infty$ as $t \to + \infty$ , there is $T$ with $τ (t) > S$ for $t > T$ , so $∥ F (τ (t)) ψ - y ∥ < ϵ$ for $t > T$ . Conversely, if $\lim_{t \to + \infty} F (τ (t)) ψ = y$ , let $ϵ > 0$ and $T$ with $∥ F (τ (t)) ψ - y ∥ < ϵ$ for $t > T$ ; for $τ > τ (T)$ the inverse $τ \mapsto 2 m τ / ℏ$ gives $t (τ) > T$ , so $∥ F (τ) ψ - y ∥ < ϵ$ . The case $t, τ \to - \infty$ is identical with reversed inequalities.

2.1F5step 1.3algebra

For a complex simple function $g = \sum_{k} a_{k} 1_{B_{k}}$ , $Φ_{F} (g) = \sum_{k} a_{k} E_{A} (B_{k} / c) = Φ_{E_{A}} (g \circ m_{c})$ , since $1_{B_{k} / c} (λ) = 1_{B_{k}} (c λ)$ . For bounded Borel $g$ , choose simple $g_{n}$ uniformly approaching $g$ ; then $g_{n} \circ m_{c}$ approaches $g \circ m_{c}$ uniformly, and [F5] gives $Φ_{F} (g) = Φ_{E_{A}} (g \circ m_{c})$ by operator-norm limits. In particular $∥ Φ_{F} (g) x ∥^{2} = \int ∣ g ∣^{2} d F_{x} = \int ∣ g (c λ) ∣^{2} d E_{A, x} (λ)$ for every $x$ .

3.1F4F5F7step 2.1algebra

Let $g_{n} (λ) : = λ 1_{{∣ λ ∣ \leq n}}$ . Step 2.1 gives $Φ_{F} (g_{n}) = Φ_{E_{A}} (c λ 1_{{∣ c λ ∣ \leq n}})$ ; the cutoff is $∣ c λ ∣ \leq n$ , not $∣ λ ∣ \leq n$ . Its squared-norm identity and [F7] applied to the increasing squared cutoffs give $\int λ^{2} d F_{x} = c^{2} \int λ^{2} d E_{A, x}$ , including possibly infinite values, for every $x$ . Thus the identity-integral domain for $F$ is exactly $D (A)$ by [F4], [F5]. On that domain, the bounded functions $k_{n} (λ) : = λ 1_{{∣ c λ ∣ \leq n}}$ approach $λ$ pointwise and satisfy $∣ k_{n} ∣ \leq ∣ λ ∣$ , so the actual cutoff-convergence clause of [F5] gives $Φ_{E_{A}} (k_{n}) x \to A x$ . Therefore $\int λ d F x = \lim_{n} Φ_{F} (g_{n}) x = c \lim_{n} Φ_{E_{A}} (k_{n}) x = c A x$ . The domain and action both match $c A$ .

4.1F4F3step 1.2step 1.3step 3.1algebra

By steps 1.3 and 3.1, the identity integral of the regular PVM $F$ equals $c A = H$ on exactly $D (A)$ . The converse and uniqueness clauses of [F4] therefore give $E_{H} = F$ , namely $E_{H} (B) = E_{A} (B / c)$ , equivalently $E_{H} (c B) = E_{A} (B)$ . Running the same argument for the self-adjoint free operator $A_{0}$ gives $E_{H_{0}} (c B) = E_{A_{0}} (B)$ . Scalar pairings yield $μ_{ψ}^{H} (c B) = μ_{ψ}^{A} (B)$ , and [F3] clause (b) gives $σ (H) = c σ (A)$ and $σ (H_{0}) = c σ (A_{0})$ , proving (b).

5.1F7step 1.2step 4.1algebra

By [F7], $m_{c}$ and its inverse preserve Lebesgue null sets. Step 4.1 transports each scalar measure by this bijection, so a measure vanishes on every null set exactly when its counterpart does; a measure supported on a null set transports to one supported on the scaled null set, and conversely. Point masses satisfy $μ_{ψ}^{H} ({λ}) = μ_{ψ}^{A} ({λ / c})$ , so absence of atoms is preserved as well. On the common domain, $H ψ = c A ψ$ makes $H ψ = λ ψ$ equivalent to $A ψ = (λ / c) ψ$ for $ψ \neq 0$ , proving (e).

5.2F6step 2.1step 4.1algebra

For $t \in R$ put $τ = ℏ t / (2 m)$ , so that $t λ / ℏ = τ μ$ with $λ = c μ$ ; by [F6] and the bounded case of 2.1 applied to the bounded measurable function $g (λ) = e^{- i t λ / ℏ} = e^{- i τ λ / c}$ one has $e^{- i t H / ℏ} = \int e^{- i t λ / ℏ} d E_{H} (λ) = \int e^{- i τ μ} d E_{A} (μ) = e^{- i τ A}$ , and the same identity with $H_{0}, A_{0}$ gives the second display of (c).

6.1step 5.2step 1.4given∎

With $F (τ) : = e^{i τ A} e^{- i τ A_{0}}$ and $G (t) : = e^{i t H / ℏ} e^{- i t H_{0} / ℏ}$ , step 5.2 gives $G (t) = F (τ (t))$ for every $t$ ; so step 1.4 shows that the strong limits ${s-lim}_{t \to \pm \infty} G (t)$ and ${s-lim}_{τ \to \pm \infty} F (τ)$ exist exactly together and then coincide. This is (d) and completes the proof.

Remarks

Use. The lemma is the exact bookkeeping that transfers every statement proved for the nondimensional operators $- Δ + M_{W}$ and their free wave operators into the physical operators $- \frac{ℏ^{2}}{2 m} Δ + M_{V}$ and their physical-time wave operators, and it carries the free spectral density along the energy rescaling $E = c λ$ . It asserts nothing about the existence of the wave operators themselves; that is the content of the scattering existence theorem of the pair.
Choice. The spectral theorem and the projection valued integrals used here carry the Axiom of Countable Choice and the Axiom of Choice; no further choice principle is used.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The three-dimensional continuity identity, the vanishing stationary divergence and the incident and outgoing current readings

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $m, ℏ > 0$ , let $Ω \subseteq R^{3}$ and $I \subseteq R$ be nonempty open sets, and let $u, v : Ω \times I \to R$ be functions of class $C^{2}$ ( $C^{k}$ Euclidean maps and diffeomorphisms) that satisfy the two real equations $ℏ \partial_{t} u = - \frac{ℏ^{2}}{2 m} Δ v + V v, - ℏ \partial_{t} v = - \frac{ℏ^{2}}{2 m} Δ u + V u$ pointwise on $Ω \times I$ for a real-valued function $V$ on $Ω \times I$ , where $x = (x_{0}, x_{1}, x_{2})$ , $\partial_{i}$ differentiates in spatial coordinate $x_{i}$ for $i \in {0, 1, 2}$ with $t$ fixed, $\partial_{t}$ differentiates in $t$ with $x$ fixed, and $Δ = \partial_{0}^{2} + \partial_{1}^{2} + \partial_{2}^{2}$ (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ). The equations are the componentwise reading of $i ℏ \partial_{t} ψ = - \frac{ℏ^{2}}{2 m} Δ ψ + V ψ$ for $ψ : = u + i v$ . Put $n : = u^{2} + v^{2} : Ω \times I \to R, j : = \frac{ℏ}{m} (u \nabla v - v \nabla u) : Ω \times I \to R^{3},$ with $\nabla$ the spatial gradient $(\partial_{0}, \partial_{1}, \partial_{2})$ of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case at each fixed $t$ and $div$ the divergence of Divergence and curl of a $C^{1}$ vector field. Then:

(a) Continuity identity. $\partial_{t} n + div j = 0$ pointwise on $Ω \times I$ .

(b) Stationary solutions. If $u, v : Ω \to R$ are $C^{2}$ and independent of $t$ , if $V : Ω \to R$ is real, and if there is a real $E$ with $- \frac{ℏ^{2}}{2 m} Δ u + V u = E u, - \frac{ℏ^{2}}{2 m} Δ v + V v = E v$ pointwise on $Ω$ , then $div j = 0$ on $Ω$ for the time-independent field $j = \frac{ℏ}{m} (u \nabla v - v \nabla u)$ , and for every closed ball $\overline{B (a, r)} \subseteq Ω$ with outward unit normal $ν$ on the sphere $\partial B (a, r)$ , $\int_{\partial B (a, r)} j \cdot ν d S = 0.$

(c) The two explicit readings. Let $k > 0$ , $d \in S^{2}$ and define $u (x) : = \cos (k d \cdot x)$ , $v (x) : = \sin (k d \cdot x)$ on $Ω = R^{3}$ (then $u, v$ are $C^{\infty}$ and satisfy the stationary equations with $V = 0$ , $E = ℏ^{2} k^{2} / (2 m)$ ). Then $j = \frac{ℏ k}{m} d on R^{3} .$ Let $r_{0} > 0$ , write $Ω : = {x \in R^{3} : ∣ x ∣ > r_{0}}$ , let $Ω ∋ x = r ω$ with $r = ∣ x ∣$ , $ω \in S^{2}$ , and let $a, b : S^{2} \to R$ be functions with $a, b \in C^{1} (S^{2})$ . Here $S^{2} : = {ω \in R^{3} : ∣ ω ∣ = 1}$ , and, for a function $g : S^{2} \to R$ , $g \in C^{1} (S^{2})$ means that its radial extension $\tilde{g} : R^{3} ∖ {0} \to R$ , $\tilde{g} (x) : = g (x / ∣ x ∣)$ , is $C^{1}$ in the open-domain sense of $C^{k}$ Euclidean maps and diffeomorphisms. Define $u (x) : = \frac{a (ω) \cos (k r) - b (ω) \sin (k r)}{r}, v (x) : = \frac{a (ω) \sin (k r) + b (ω) \cos (k r)}{r}$ on $Ω$ , so that $u + i v = \frac{f (ω)}{r} e^{i k r}$ with $f : = a + i b$ . Then $u, v \in C^{1} (Ω)$ and, with $\hat{r} : = ω$ , $j \cdot \hat{r} = \frac{ℏ k}{m} \frac{∣ f (ω) ∣^{2}}{r^{2}}, equivalently r^{2} j \cdot \hat{r} = \frac{ℏ k}{m} ∣ f (ω) ∣^{2},$ and the current identity holds for every declared $f \in C^{1} (S^{2})$ , with no equation required of the field.

The assertion is an identity for the declared differentiable functions; it constructs no probability law, no state, no beam and no cross section. The fields of clause (c) are only $C^{1}$ : an $f \in C^{1} (S^{2})$ need not admit two angular derivatives, and no outgoing field is claimed here to solve the free equation or to be stationary.

Facts & Assumptions

Given: the open sets $Ω, I$ , the functions $u, v, V$ , the constants $m, ℏ > 0$ , and the data of clause (c).

[F1]

Differentiation rules. For totally differentiable Euclidean maps the sum and scalar-multiple rules hold (Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives) and the chain rule $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ holds (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ), with $D f (a) e_{j} = \partial_{j} f (a)$ recovering the partial derivatives (A total derivative computes every directional derivative, and its matrix is the Jacobian); the partial derivatives themselves are those of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , the entries of the Jacobian of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case. For scalar functions on an interval the sum, product and quotient rules hold (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ), and $(\sin)^{'} = \cos$ and $(\cos)^{'} = - \sin$ on $R$ (The derivatives of sine and cosine are cosine and minus sine). Also $\sin^{2} s + \cos^{2} s = 1$ for every real $s$ (Parity and the Pythagorean identity for sine and cosine).

[F2]

Divergence of a product with a scalar. For $C^{1}$ functions $φ : Ω \to R$ and $F : Ω \to R^{3}$ , the equality $div (φ F) = ⟨ \nabla φ, F ⟩ + φ div F$ holds pointwise, the divergence and gradient being those of Divergence and curl of a $C^{1}$ vector field and The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case (Divergence and curl are linear and satisfy the scalar product rules).

[F3]

Divergence theorem. Assume ${A C}_{ω}$ . For a bounded $C^{1}$ domain $Ω_{0}$ and $F \in C^{1} (\overline{Ω_{0}}; R^{3})$ , $\int_{Ω_{0}} div F d x = \int_{\partial Ω_{0}} F \cdot ν d S$ (Divergence on a bounded C1 Euclidean domain).

Proof

technique · direct; differentiate $n$, use the two real equations to cancel the terms, then evaluate the stationary divergence and the two explicit fields

Given: the data of the statement and [F1]-[F3].

1.1F1given

By [F1], $\partial_{t} n = \partial_{t} (u^{2} + v^{2}) = 2 (u \partial_{t} u + v \partial_{t} v)$ pointwise on $Ω \times I$ .

1.2F2F1algebra

Put $F : = u \nabla v - v \nabla u$ . By [F2] applied to each of the two products, using $div (\nabla v) = \sum_{i = 0}^{2} \partial_{i} \partial_{i} v = Δ v$ and likewise for $u$ , one has $div (u \nabla v) = \nabla u \cdot \nabla v + u Δ v$ and $div (v \nabla u) = \nabla v \cdot \nabla u + v Δ u$ ; the mixed terms are equal by symmetry of the inner product and cancel in the difference, so $div F = u Δ v - v Δ u$ and hence $div j = \frac{ℏ}{m} (u Δ v - v Δ u)$ .

1.3givenalgebra

Reading the complex equation $i ℏ \partial_{t} (u + i v) = - \frac{ℏ^{2}}{2 m} Δ (u + i v) + V (u + i v)$ by real and imaginary parts gives $ℏ \partial_{t} u = - \frac{ℏ^{2}}{2 m} Δ v + V v$ and $- ℏ \partial_{t} v = - \frac{ℏ^{2}}{2 m} Δ u + V u$ , i.e. $\partial_{t} u = - \frac{ℏ}{2 m} Δ v + \frac{V}{ℏ} v$ and $\partial_{t} v = \frac{ℏ}{2 m} Δ u - \frac{V}{ℏ} u$ .

1.4F1givenalgebra

For $u = \cos (k d \cdot x)$ , $v = \sin (k d \cdot x)$ the chain rule [F1] gives $\nabla u = - k d \sin (k d \cdot x)$ and $\nabla v = k d \cos (k d \cdot x)$ ; hence $u \nabla v - v \nabla u = k d (\cos^{2} + \sin^{2}) = k d$ and $j = \frac{ℏ k}{m} d$ . The two functions are $C^{\infty}$ and satisfy the stationary equations with $V = 0$ , because $\partial_{i} \partial_{i} u = - k^{2} d_{i}^{2} \cos (k d \cdot x)$ and $\partial_{i} \partial_{i} v = - k^{2} d_{i}^{2} \sin (k d \cdot x)$ by [F1], so summing over $i = 0, 1, 2$ and using $∣ d ∣^{2} = 1$ gives $Δ u = - k^{2} u$ and $Δ v = - k^{2} v$ with $E = ℏ^{2} k^{2} / (2 m)$ .

1.5F1givenalgebra

Write $c_{k} : = \cos (k r)$ , $s_{k} : = \sin (k r)$ and $f = a + i b$ . Then $u = r^{- 1} (a c_{k} - b s_{k})$ , $v = r^{- 1} (a s_{k} + b c_{k})$ , so by the chain and product rules for functions of the single variable $r$ [F1], $\partial_{r} u = - k r^{- 1} (a s_{k} + b c_{k}) - r^{- 2} (a c_{k} - b s_{k})$ and $\partial_{r} v = k r^{- 1} (a c_{k} - b s_{k}) - r^{- 2} (a s_{k} + b c_{k})$ . Multiplying out, $u \partial_{r} v - v \partial_{r} u = k r^{- 2} [(a c_{k} - b s_{k})^{2} + (a s_{k} + b c_{k})^{2}] = k r^{- 2} (a^{2} + b^{2}) = k r^{- 2} ∣ f (ω) ∣^{2}$ ; since $\hat{r}$ is a unit vector and the radial directional derivative of an angular function is zero, $j \cdot \hat{r} = \frac{ℏ}{m} (u \partial_{r} v - v \partial_{r} u)$ , which is the first display of (c), and multiplying by $r^{2} > 0$ gives the equivalent second display. Both $u$ and $v$ are $C^{1}$ on $Ω$ : they are sums of products of the $C^{\infty}$ radial factors $r^{- 1}$ , $\cos (k r)$ , $\sin (k r)$ with the $C^{1}$ angular factors $a, b$ , and the radial extensions of $a, b$ are $C^{1}$ by the explicit convention in (c), so the Euclidean product rules give the stated regularity.

2.1step 1.1step 1.2step 1.3algebra

Substituting 1.3 into 1.1 and using that $V$ is real, $2 (u \partial_{t} u + v \partial_{t} v) = 2 (- \frac{ℏ}{2 m} u Δ v + \frac{V}{ℏ} u v + \frac{ℏ}{2 m} v Δ u - \frac{V}{ℏ} v u) = - \frac{ℏ}{m} (u Δ v - v Δ u)$ . Adding the value of $div j$ from 1.2 gives $\partial_{t} n + div j = 0$ , which is (a).

2.2step 1.2givenalgebra

If $u, v, V$ do not depend on $t$ and satisfy the stationary equations, then $Δ u = \frac{2 m}{ℏ^{2}} (V - E) u$ and $Δ v = \frac{2 m}{ℏ^{2}} (V - E) v$ pointwise, so 1.2 gives $div j = \frac{ℏ}{m} \cdot \frac{2 m}{ℏ^{2}} (V - E) (u v - v u) = 0$ on $Ω$ .

3.1F3step 1.4step 1.5step 2.2algebra∎

Let $\overline{B (a, r)} \subseteq Ω$ . The ball $B (a, r)$ is a bounded $C^{1}$ domain and $j$ is $C^{1}$ on a neighbourhood of its closure, so [F3] gives $\int_{\partial B (a, r)} j \cdot ν d S = \int_{B (a, r)} div j d x = 0$ by 2.2; with 1.4 for the plane-wave reading, 1.5 for the outgoing current identity, 2.1 for (a) and this step for (b), the proof of (a), (b) and (c) is complete.

Remarks

Role. The lemma supplies the exact current reading used by the pair's stationary beam model: $j$ is the pair of clauses (a) and (b) that makes the current of a stationary solution divergence-free and flux-free through spheres, and clause (c) computes the two currents -- the incident plane wave with magnitude $ℏ k / m$ and the outgoing spherical wave with radial density $(ℏ k / m) ∣ f ∣^{2} / r^{2}$ -- that the model's cross-section reading compares. Clause (c) asserts the outgoing current identity without any equation for the field: the model's amplitude $f_{k}$ appears in the outgoing coefficient of the full stationary solution (incident wave plus outgoing wave plus remainder), whose existence and equation are established by the pair's stationary theorem, not here.
Choice. The divergence theorem used in 3.1 assumes countable choice; no additional choice principle is used.

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The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units

Physical scope

The declared short-range scattering model of the pair: the spin-zero massive one-particle state space, an assigned external real potential in one of two declared short-range classes (bounded compact support, or a square-integrable plus bounded decomposition whose shifted free resolvent tail is integrable), the free and interacting assigned energy observables with the Coulomb tail explicitly excluded from the short-range classes, the Born position law, and the wave-operator and asymptotic-completeness vocabulary of the pair. The model is an additional assumption of the pair with a prescribed, not derived, potential; the long-range Coulomb case is a separate declared model. No preparation, no apparatus, no detector and no measured quantity appears.

Definition

(a) Framework data. Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4, the kinematics and massive-particle model postulate P5, the identical-particle and spin postulate P6, and the preparation and apparatus discipline P7 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). Fix a mass parameter $m > 0$ and a spin label $s = 0$ , so that the framework's one-particle model The massive spin-s particle model of the framework: state space, operators, units and frame readings reads its state space as $H_{0} = L^{2} (R^{3}; C)$ with the pairing linear in the first argument, its componentwise position family $Q_{1}, Q_{2}, Q_{3}$ , its componentwise momentum family $P_{1}, P_{2}, P_{3}$ and the action constant $ℏ > 0$ ; the spin-zero internal factor carries the trivial representation of P6. Units are those of Dimensions, SI unit conventions and quantity types of the framework: position is read in metres, momentum in $k g m s^{- 1}$ , energy and potential energy in joules, and $ℏ$ in $J s$ . Assume the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the cited mathematical suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice).

Choose a reference length $L_{*} > 0$ in metres and put $y = x / L_{*}, E_{*} : = \frac{ℏ^{2}}{2 m L_{*}^{2}}, T_{*} : = ℏ / E_{*} .$ Here $y$ is dimensionless, $E_{*}$ is in joules and $T_{*}$ in seconds. The unitary coordinate dilation from physical-coordinate $L_{x}^{2}$ to $L_{y}^{2}$ is $(D_{*} f) (y) = L_{*}^{3 / 2} f (L_{*} y)$ , with inverse $(D_{*}^{*} g) (x) = L_{*}^{- 3 / 2} g (x / L_{*})$ . Lebesgue dilation (For a nonzero real $c$ , dilation by $c$ multiplies Lebesgue outer measure by $∣ c ∣^{n}$ , and reflection in the origin preserves it) gives its norm identity, first for simple functions and then by nonnegative approximation. This convention makes all coordinates and spectral shifts in the tail criterion below dimensionless; physical stationary equations retain $x$ .

(b) The declared scattering model and its potential classes. As an additional model assumption of the pair, adopt the short-range scattering model. Its declared state space is $H_{0} = L^{2} (R^{3}; C)$ of clause (a), with the position and momentum families of that clause. A declared short-range potential of the model is a real-valued measurable function $V : R^{3} \to R$ satisfying either

(compact-support class) $V$ is bounded, $∥ V ∥_{\infty} < \infty$ , and $V = 0$ almost everywhere outside some ball $B_{s u p p} : = {x : ∣ x ∣ \leq R_{s u p p}}$ with $R_{s u p p} < \infty$ ; or
(resolvent-tail class) $V = V_{2} + V_{b}$ with real $V_{2} \in L^{2} (R^{3})$ and real bounded $V_{b}$ , and, with the dimensionless reduced potential $W_{*} (y) : = V (L_{*} y) / E_{*} = W_{*, 2} (y) + W_{*, b} (y)$ and $A_{0, *} : = - Δ_{y}^{w e a k}$ on $W^{2, 2} (R_{y}^{3})$ , the function $h_{*} (r) : = ∥ M_{W_{*}} (A_{0, *} - i I)^{- 1} 1_{{∣ y ∣ \geq r}} ∥, r \geq 0,$ is finite for every dimensionless $r$ and belongs to $L^{1} ([0, \infty), d r)$ . Equivalently the physical-coordinate operator is $M_{V} (H^{f r e e} - i E_{*} I)^{- 1} 1_{{∣ x ∣ \geq L_{*} r}}$ : conjugation by $D_{*}$ gives exactly the displayed operator. Thus physical cutoffs have radius $L_{*} r$ , and integration in physical radius $R$ gives $\int_{0}^{\infty} h_{*} (R / L_{*}) d R = L_{*} \int_{0}^{\infty} h_{*} (r) d r$ . The reference length is fixed as part of this unit convention.

The function $V$ is read as a potential energy in joules; the model asserts no derivation of $V$ from P5 or from any force law, and an assignment of a concrete target to $V$ is a tested identification in the sense of clause (iii) of P7 carrying the corresponding uncertainty. The pure Coulomb potential $V (x) = - κ / ∣ x ∣$ , $κ > 0$ , is not in either class: it is a declared long-range model with its own definition and comparison construction on this page.

(c) Assigned energy observables. Let $F_{2}$ be the unitary Plancherel transform of Real multiplication operators and free plus bounded-potential Schrödinger operators clause (b), $c : = 2 π^{2} ℏ^{2} / m$ , and let $H^{f r e e} : = F_{2}^{*} M_{c ∣ ξ ∣^{2}} F_{2}, D (H^{f r e e}) : = {f \in H_{0} : ∣ ξ ∣^{2} F_{2} f \in H_{0}}$ be the free operator of that clause with $d = 3$ and $b = 0$ ; by The three-dimensional free Schrödinger operator as the weak Laplacian on its Sobolev domain its domain is the Sobolev space $W^{2, 2} (R^{3}; C)$ (Integer-order Sobolev spaces and their norms) and $H^{f r e e} f = - \frac{ℏ^{2}}{2 m} Δ_{w} f$ with $Δ_{w}$ the weak Laplacian (Weak derivative of a locally integrable function). For equations in physical coordinates write $A_{0, x} : = - Δ_{x}^{w e a k}$ and $W_{x} (x) : = 2 m V (x) / ℏ^{2}$ , so $H^{f r e e} = (ℏ^{2} / 2 m) A_{0, x}$ . Both $A_{0, x}$ and $W_{x}$ have inverse-square-length units; this energy division alone is not a spatial nondimensionalization. Stationary items may abbreviate $W_{x}$ as $W$ (or its central radial profile as $w$ ), keeping physical $k$ and $r$ . The dimensionless operators instead satisfy $D_{*} H^{f r e e} D_{*}^{*} = E_{*} A_{0, *}, D_{*} H D_{*}^{*} = E_{*} (A_{0, *} + M_{W_{*}}), W_{*} (y) = L_{*}^{2} W_{x} (L_{*} y) .$ Indeed testing the weak derivative definition after $x = L_{*} y$ gives $\partial_{y}^{α} D_{*} f = L_{*}^{∣ α ∣} D_{*} (\partial_{x}^{α} f)$ for $∣ α ∣ \leq 2$ ; applying the inverse dilation gives the converse, so $D_{*}$ maps the physical Sobolev domain onto the dimensionless one. Multiplication transforms by direct substitution, proving both operator identities on their exact domains. The subscript $w$ in $Δ_{w}$ above denotes weak derivatives, not a coordinate or a unit convention. With $M_{V}$ the maximal multiplication operator, $D (M_{V}) = {f \in L^{2} : V f \in L^{2}}$ (bounded and everywhere defined in the compact-support class), the assigned energy observables of the model with declared potential $V$ are the free operator $H^{f r e e}$ and the interacting operator $H : = H^{f r e e} + M_{V}, D (H) : = D (H^{f r e e}) .$ The declared energy unit is the joule. For the compact-support class, clause (c) of the cited theorem makes $H$ self-adjoint on $D (H)$ and bounded below by $- ∥ V ∥_{\infty}$ . For the resolvent-tail class, Resolvent-tail short-range completeness (Statement and proof step 1.1) applies to $W_{*} = W_{*, 2} + W_{*, b}$ : $∥ W_{*, 2} ∥_{2} = E_{*}^{- 1} L_{*}^{- 3 / 2} ∥ V_{2} ∥_{2} < \infty$ , $∥ W_{*, b} ∥_{\infty} = E_{*}^{- 1} ∥ V_{b} ∥_{\infty} < \infty$ , and its tail hypothesis is exactly clause (b)2. That supplier proves $H_{y}^{2} \subseteq D (M_{W_{*}})$ and self-adjointness of $A_{0, *} + M_{W_{*}}$ on $H_{y}^{2} = W_{y}^{2, 2}$ . The displayed unitary dilation and positive scaling therefore give $D (H^{f r e e}) \subseteq D (M_{V})$ and self-adjointness of $H$ on precisely $W_{x}^{2, 2}$ , without assuming that $V_{2}$ is bounded. The model adopts $(H^{f r e e}, J)$ and $(H, J)$ as assigned sharp observables in the sense of Assigned sharp observables and the outcome law of the framework clause (a), and reads their flows by Assigned energy observable, unitary flow and the two pictures of autonomous evolution with $U_{f r e e} (t) = e^{- i t H^{f r e e} / ℏ}$ and $U (t) = e^{- i t H / ℏ}$ .

(d) Short-range scattering vocabulary. For a declared short-range potential $V$ the model declares the free asymptotic comparison to be the pair of strong limits $Ω_{\pm} : = {s-lim}_{t \to \pm \infty} e^{i t H / ℏ} e^{- i t H^{f r e e} / ℏ},$ which are declared wave operators of the model when both limits exist, and declares the scattering operator $S : = Ω_{+}^{*} Ω_{-}$ when both exist and $Ω_{\pm}$ are isometries. The model says that the pair is asymptotically complete for this $V$ when the two ranges agree with the subspace $H_{a c} (H) : = {ψ \in H_{0} : μ_{ψ}^{H} ≪ λ},$ where $μ_{ψ}^{H} (B) : = ⟨ E_{H} (B) ψ, ψ ⟩$ is the scalar spectral measure of the projection valued measure $E_{H}$ of the self-adjoint operator $H$ , and with $H_{c} (H) : = H_{p p} (H)^{⊥}, H_{p p} (H) : = \overset{}{span} {ψ \neq 0 : H ψ = λ ψ for some λ \in R};$ here $H_{p p} (H)$ is the closed linear span of the eigenvectors of $H$ . An asymptotic free state of a vector $ψ \in H_{0}$ is a vector $ψ_{i n}$ such that $∥ e^{- i t H / ℏ} ψ - e^{- i t H^{f r e e} / ℏ} ψ_{i n} ∥ \to 0$ as $t \to - \infty$ ; the model reads this limit relation through the wave-operator identity $ψ = Ω_{-} ψ_{i n}$ only when the wave operator exists. These are declarations of the model under P4; they construct no preparation, detector or measured count, and no definition of a cross section appears here.

(e) Status and scope. Clause (a) adopts P1, P2, P4, P5, P6 and P7 and fixes the pair's spin-zero one-particle data, mass, action constant and choice hypotheses. Clause (b) further adopts the compact-support and resolvent-tail potential classes as additional pair-level model assumptions; neither class follows from those postulates or from the cited mathematical results. Clause (c) declares the assigned energy operators and their common domain; the cited mathematics establishes self-adjointness where stated, including the resolvent-tail case supplied by the cited mathematical theorem. Clause (d) declares the comparison and spectral vocabulary without asserting that the limits exist or are complete. No identification of $m$ , $ℏ$ or $V$ with measured data is asserted. Existence of the limits, completeness of the ranges and every cross section are separate results of the pair.

Remarks

Why two classes. The compact-support class is the minimal physically transparent case in which the pair's completeness theorem is proved; the resolvent-tail class is the wider class in which the same conclusion is proved from an integrable decay hypothesis on the shifted free resolvent tail. The Coulomb potential fails both classes and is treated separately by the long-range definition of this page.
Names. The symbols $Ω_{\pm}$ , $S$ , $H_{a c}$ , $H_{p p}$ and $H_{c}$ are declared here once and used with exactly this meaning on this page and its companion.

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The declared stationary elastic beam model: incident plane wave, scattered wave, current readings and differential and total cross sections

Physical scope

The pair's declared stationary elastic beam model for the short-range scattering model: a monochromatic incident plane-wave reading with wave number and direction, the corresponding current density, the scattered outgoing-wave reading, the declared generalized stationary solution, the idealization that the stationary wave is not a state-space element, the small-target/elastic equal-channel flux assumptions under which the differential and total cross sections are read as ratios of currents, and the units of those quantities. The model is an additional assumption of the pair; it asserts no preparation, no beam apparatus, no count rate and no measured cross section, and no single-particle trajectory claim.

Definition

(a) Framework data. Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 and the preparation and apparatus discipline P7 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), with the short-range scattering model of The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units: mass $m > 0$ , spin zero, state space $H_{0} = L^{2} (R^{3}; C)$ , a declared short-range potential $V$ , the free and interacting assigned energy observables $H^{f r e e}$ and $H$ , its reading of the flow by Assigned energy observable, unitary flow and the two pictures of autonomous evolution, and the units of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice).

(b) Incident plane-wave reading and declared density scale. As an additional model assumption of the pair, adopt the stationary elastic beam model. Fix a wave number $k > 0$ in $m^{- 1}$ , a direction $d \in S^{2}$ , and an incident number-density scale $n_{i n c} > 0$ in $m^{- 3}$ . This scale is an explicitly declared model normalization; no numerical sample density or measured beam value is supplied. Retain the unit-amplitude, dimensionless incident shape $u_{i n c} (x) : = e^{i k d \cdot x},$ with energy reading $E : = ℏ^{2} k^{2} / (2 m)$ in joules. The physical incident beam field is $Φ_{i n c} : = \sqrt{n_{i n c}} u_{i n c}$ , with unit $m^{- 3 / 2}$ and constant model number density $∣ Φ_{i n c} ∣^{2} = n_{i n c}$ . For a dimensionless unit-amplitude shape $ψ$ , write $j^{u n i t} [ψ] : = (ℏ / m) Im (\overline{ψ} \nabla ψ)$ for the mathematical current of The three-dimensional continuity identity, the vanishing stationary divergence and the incident and outgoing current readings. Its incident value is $(ℏ k / m) d$ , with velocity unit $m s^{- 1}$ . The model's physical number-current density is the current of $Φ = \sqrt{n_{i n c}} ψ$ , $j [Φ] : = \frac{ℏ}{m} Im (\overline{Φ} \nabla Φ) = n_{i n c} j^{u n i t} [ψ] .$ Thus the incident current density and incident flux density through a plane normal to $d$ are $j_{i n c} = n_{i n c} \frac{ℏ k}{m} d, J_{i n c} : = n_{i n c} \frac{ℏ k}{m} > 0,$ both in $m^{- 2} s^{- 1}$ as in Dimensions, SI unit conventions and quantity types of the framework clause (d)5. The mathematical plane-wave current of the cited lemma remains the unit-shape computation; the density factor supplies the declared physical beam normalization.

(c) Scattered-wave reading and generalized stationary solution. For a declared short-range potential $V$ the model declares a generalized stationary solution at energy $E$ and direction $d$ to be a function $Ψ : R^{3} \to C$ in $H_{l o c}^{2} (R^{3})$ , meaning that all weak derivatives of order at most two are locally square integrable (Integer-order Sobolev spaces and their norms), with $V Ψ \in L_{l o c}^{2}$ , of class $C^{2}$ on the complement of the essential closed support of $V$ , and satisfying the equation on all of $R^{3}$ distributionally: $\int_{R^{3}} Ψ (x) [- \frac{ℏ^{2}}{2 m} Δ η (x) + (V (x) - E) η (x)] d x = 0 (η \in C_{c}^{\infty} (R^{3}; C)) .$ Here the essential closed support is the complement of the union of open sets on which $V = 0$ almost everywhere; the test functions are smooth and compactly supported; $Δ = \sum_{i = 0}^{2} \partial_{i}^{2}$ uses the spatial coordinate convention of the current lemma. Outside the essential closed support, $V = 0$ almost everywhere and $Ψ$ is $C^{2}$ , so the continuous free-equation residual has zero test pairing and hence vanishes pointwise: $- \frac{ℏ^{2}}{2 m} Δ Ψ = E Ψ$ . The equation containing the declared representative $V (x)$ holds almost everywhere there; it holds pointwise there when $V$ is replaced by its zero representative on this exterior open set. No pointwise equation with arbitrary null-set values of $V$ is asserted. The solution has the outgoing asymptotic reading $Ψ (x) = e^{i k d \cdot x} + \frac{f_{k} (Ω)}{r} e^{i k r} + E (x), x = r Ω, r \to \infty,$ with a scattering amplitude function $f_{k} : S^{2} \to C$ , read in metres relative to the unit incident shape, and a physical beam field $Φ : = \sqrt{n_{i n c}} Ψ$ with unit $m^{- 3 / 2}$ . The same constant factor multiplies its incident, outgoing and remainder terms; $f_{k}$ remains the amplitude relative to the incident field, in metres. Here $Ω : = x / r$ is the outgoing unit direction. Require $f_{k} \in C^{1} (S^{2})$ in the radial-extension convention of The three-dimensional continuity identity, the vanishing stationary divergence and the incident and outgoing current readings (c), and require $∣ E (r, Ω) ∣ + ∣ \partial_{r} E (r, Ω) ∣ \leq C r^{- 2}$ uniformly in $Ω$ for all sufficiently large $r$ , with some finite $C$ . For the bounded compact-support class the model selects the outgoing integral solution of Stationary outgoing integral and flux, with its potential parameter replaced by $W_{x} = 2 m V / ℏ^{2}$ . Explicitly, choose $R > 0$ enclosing the essential support, put $G_{k} (z) = e^{i k ∣ z ∣} / (4 π ∣ z ∣)$ for $z \neq 0$ , and on $L^{2} (B_{R}; C)$ put $K_{k} u (x) = \int_{B_{R}} G_{k} (x - y) W_{x} (y) u (y) d y$ . Select $u = (I + K_{k})^{- 1} (u_{i n c} ∣_{B_{R}})$ and $Ψ (x) = u_{i n c} (x) - \int_{B_{R}} G_{k} (x - y) W_{x} (y) u (y) d y$ , using the locally continuous representative supplied by $H_{l o c}^{2}$ . The cited theorem proves invertibility and the global equation, with a smooth amplitude and these uniform remainder bounds. The selection is independent of enlarging $R$ : the restriction of a solution constructed in a larger ball solves the smaller invertible integral equation, and $W_{x} = 0$ outside the smaller ball makes both global integrals identical. Hence the compact-support model determines $Ψ$ and $f_{k}$ from $V, k, d$ ; arbitrary exterior fields and arbitrary interior extensions are excluded. For the general resolvent-tail class these requirements define a conditional reading only, without an existence or uniqueness assertion. The scattered physical field is $Φ_{s c} : = \sqrt{n_{i n c}} (Ψ - u_{i n c})$ . The physical scattered radial current density, obtained by applying the quadratic current formula to this scattered component, is $j_{r}^{s c} (r, Ω) = n_{i n c} (\frac{ℏ k}{m} \frac{∣ f_{k} (Ω) ∣^{2}}{r^{2}} + O (r^{- 3})) (r \to \infty),$ in $m^{- 2} s^{- 1}$ for the fixed declared density scale. The leading unit-shape current is clause (c) of The three-dimensional continuity identity, the vanishing stationary divergence and the incident and outgoing current readings; the current of the remainder is controlled by the stationary theorem. The definition uses the scattered component, with no assumption that incident/scattered interference in the total field vanishes. A generalized stationary solution is generally not an element of $H_{0}$ ; it is not a preparation of the model, and clause (d) of The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units does not apply to it. It is used only through the declared flux quantities computed from it, and every such use states the idealization.

(d) Differential and total cross-section readings. Under the declared equal-channel elastic flux assumptions -- that the incident and outgoing readings refer to the same elastic channel at the same energy, that the incident flux density is spatially uniform over the transverse extent of the target and the outgoing current is read far outside the support of $V$ -- the model declares the differential cross section $\frac{d σ}{d Ω} (Ω) : = \lim_{r \to \infty} \frac{r^{2} j_{r}^{s c} (r, Ω)}{J_{i n c}} = ∣ f_{k} (Ω) ∣^{2},$ read in $m^{2} s r^{- 1}$ , and the total cross section $σ : = \int_{S^{2}} \frac{d σ}{d Ω} (Ω) d Ω = \int_{S^{2}} ∣ f_{k} (Ω) ∣^{2} d Ω,$ read in $m^{2}$ , whenever the displayed limit and integral exist; the existence and finiteness of both are proved on this page for the bounded compact-support class at $k > 0$ . No stationary existence or finite cross-section result is asserted here for the general resolvent-tail class. The common positive density factor cancels in the ratio, $r^{2} j_{r}^{s c} / J_{i n c} \to ∣ f_{k} ∣^{2}$ , so the cross sections are independent of the declared $n_{i n c}$ . This normalization supplies no relation to the energy outcome law of P2 and no assertion that an $L^{2}$ preparation realizes either stationary field.

(e) Status and scope. Clauses (b)-(d) are declarations of the pair under P1, P2, P4 and P7 together with the short-range model; they are an idealized stationary reading and not a model of an apparatus. The new density scale is a symbolic normalization parameter of that reading, not a measured sample value; multiplication by $\sqrt{n_{i n c}}$ does not turn the stationary shape into an $L^{2}$ preparation. In particular no beam source, target density, collimation, detector acceptance, efficiency, background or counting procedure appears, and the symbols $d σ / d Ω$ and $σ$ denote model quantities computed from the declared solution. An identification of a measured count cross section with these quantities requires a separate preparation and detection model and carries the uncertainty of that identification.

Remarks

Flux, not probability. The cross-section readings are ratios of current densities of the declared physical stationary fields, whose currents are $n_{i n c}$ times the unit-shape currents computed in The three-dimensional continuity identity, the vanishing stationary divergence and the incident and outgoing current readings; they are not read as probabilities of detecting a single particle, and no such single-particle statement is asserted.
Optical theorem and other refinements. No optical theorem, dispersion relation or inelastic extension is declared or used on this pair.

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Existence of the model's wave operators, isometry and intertwining, and the purely absolutely continuous free spectrum

Physical scope

The declared short-range scattering model with a real bounded compactly supported potential, for the assigned free and interacting energy observables of the pair: self-adjointness of the interacting operator on the Sobolev domain, the Cook integrability of the potential along the free evolution, existence of the model's wave operators as strong limits with isometry and intertwining, the purely absolutely continuous free spectrum with the explicit free spectral density in the pair's variables, and the absence of positive eigenvalues of the interacting operator. Conditional on P1, P2, P4, P5, P6, P7, the declared model and the cited mathematics; no measurement, detector, cross section or empirical value is asserted.

Statement

Adopt the declared short-range scattering model The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units with mass parameter $m > 0$ , spin zero, action constant $ℏ > 0$ , and a declared potential $V$ of the compact-support class, so that $V$ is real, bounded and vanishes almost everywhere outside a ball $B_{R}$ ; let $H^{f r e e}$ and $H = H^{f r e e} + M_{V}$ be the assigned energy observables of the model on $D (H) = W^{2, 2} (R^{3}; C)$ , and let $Ω_{\pm}$ , $S$ , $μ_{ψ}^{H}$ and $H_{a c} (H)$ be as in clause (d) of that definition. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Then:

(a) $H$ is self-adjoint on $D (H) = W^{2, 2} (R^{3}; C)$ and bounded below, and for every Schwartz function $ψ \in S (R^{3}; C)$ (Schwartz space and its seminorms) $\int_{0}^{\infty} ∥ V e^{- i t H^{f r e e} / ℏ} ψ ∥_{2} d t < \infty, \int_{- \infty}^{0} ∥ V e^{- i t H^{f r e e} / ℏ} ψ ∥_{2} d t < \infty;$

(b) the strong limits $Ω_{\pm} = {s-lim}_{t \to \pm \infty} e^{i t H / ℏ} e^{- i t H^{f r e e} / ℏ}$ exist as bounded linear isometries of $H_{0}$ , so $Ω_{\pm}^{*} Ω_{\pm} = I$ ; they intertwine the two flows, $e^{- i s H / ℏ} Ω_{\pm} = Ω_{\pm} e^{- i s H^{f r e e} / ℏ} (s \in R),$ and satisfy $H Ω_{\pm} ψ = Ω_{\pm} H^{f r e e} ψ$ for every $ψ \in D (H^{f r e e})$ ;

(c) the free operator has purely absolutely continuous spectrum: for every $ψ \in H_{0}$ the scalar measure $μ_{ψ}^{H^{f r e e}}$ is absolutely continuous with respect to Lebesgue measure and gives zero mass to ${0}$ and to $(- \infty, 0)$ ; hence $H_{a c} (H^{f r e e}) = H_{c} (H^{f r e e}) = H_{0}$ and $H^{f r e e}$ has no eigenvalues;

(d) $H$ has no eigenvalue in $(0, \infty)$ : if $ψ \in D (H)$ , $ψ \neq 0$ and $H ψ = λ ψ$ with $λ > 0$ , then no such $ψ$ exists.

The statement is conditional on the declared model and the cited mathematics. It asserts no completeness of the ranges, no unitarity of $S$ , no cross section and no measured quantity; completeness is the content of the two following theorems of this page.

Facts & Assumptions

Given: the declared short-range scattering model with $m, ℏ > 0$ , spin zero and a compact-support-class potential $V$ with support in $B_{R}$ ; the operators $H^{f r e e}$ , $H$ on $W^{2, 2}$ ; the choice hypotheses.

[F1]

Scaling correspondence. With $c = ℏ^{2} / (2 m)$ and $W : = 2 m V / ℏ^{2}$ , the operators $A_{0} : = - Δ_{w}$ and $A : = - Δ_{w} + M_{W}$ on $W^{2, 2} (R^{3}; C)$ satisfy $H^{f r e e} = c A_{0}$ and $H = c A$ on that domain, $E_{H} (c B) = E_{A} (B)$ for every Borel $B$ , $e^{- i t H / ℏ} = e^{- i τ A}$ and $e^{- i t H^{f r e e} / ℏ} = e^{- i τ A_{0}}$ with $τ = ℏ t / (2 m)$ , the strong limits in the physical and energy-divided times coincide, and $H$ has an eigenvalue $λ > 0$ if and only if $A$ has the eigenvalue $λ / c > 0$ (Unit, time and spectral correspondence between the physical scattering operators and the nondimensional Laplacian clauses (a), (b), (c), (d), (e)).

[F2]

Concrete Cook existence. Assume Countable Choice. For real $V_{0} \in L^{2} (R^{3})$ , with $H_{0} = - Δ$ on $W^{2, 2}$ and $H = H_{0} + M_{V_{0}}$ self-adjoint on the same domain, the Cook integrals $\int_{0}^{\infty} ∥ V_{0} e^{- i t H_{0}} ψ ∥_{2} d t$ and its negative-time analogue converge for every Schwartz $ψ$ , the wave operators $Ω_{\pm} = {s-lim}_{t \to \pm \infty} e^{i t H} e^{- i t H_{0}}$ exist as bounded linear isometries with $Ω_{\pm} e^{- i s H_{0}} = e^{- i s H} Ω_{\pm}$ and $H Ω_{\pm} ψ = Ω_{\pm} H_{0} ψ$ for $ψ \in D (H_{0})$ , and $H_{0}$ has purely absolutely continuous spectrum with $L^{2} = H_{a c} (H_{0})$ , the scalar spectral measure of $ψ$ having the displayed density with respect to Lebesgue measure and zero mass at $0$ (Concrete three-dimensional Cook existence clauses (a), (b), (c)).

[F3]

Cook isometry and intertwining. Under the hypotheses of [F2] the wave operators satisfy $Ω_{\pm}^{*} Ω_{\pm} = I$ , the intertwining identity, and $Ω_{\pm} D (H_{0}) \subseteq D (H)$ with $H Ω_{\pm} ψ = Ω_{\pm} H_{0} ψ$ ; no surjectivity is asserted (Cook isometry and intertwining clauses (a)-(d)).

[F4]

No positive compact-potential eigenvalues. Assume Countable Choice. For real bounded compactly supported $V_{0}$ and $k > 0$ , the operator $- Δ + M_{V_{0}}$ on $H^{2} (R^{3})$ has no eigenvalue $k^{2}$ (No positive compact-potential eigenvalues).

[F5]

The model's readings. The assigned energy observables are $(H^{f r e e}, J)$ and $(H, J)$ on $H_{0} = L^{2} (R^{3}; C)$ with flow $U (t) = e^{- i t H / ℏ}$ ; in the compact-support class $H$ is self-adjoint on $D (H)$ and bounded below by $- ∥ V ∥_{\infty}$ , and the wave operators and the scattering operator are read as the strong limits of the model's flow comparison (The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units clauses (c), (d)).

Proof

technique · direct; use physical-coordinate $W=W_x$ and $A_0=A_{0,x}$, whose numerical representatives in the fixed metre convention are the abstract source operators; the auxiliary parameter $\tau=\hbar t/(2m)$ has square-metre units; identify the model's operators with the energy-divided operators of the cited mathematics through the scaling lemma, transfer the Cook estimates by the linear change of variables $\tau=\hbar t/(2m)$, and transfer the spectral statements by the projection valued measure correspondence

Given: the data and hypotheses of the statement and [F1]-[F5].

1.1F1F2F5given

By [F5], the model supplies self-adjointness of $H = H^{f r e e} + M_{V}$ on $W^{2, 2}$ and its lower bound $- ∥ V ∥_{\infty}$ . By [F1], the reduced operator $A = - Δ_{w} + M_{W}$ is self-adjoint on the same domain and $H = c A$ , with $W = 2 m V / ℏ^{2}$ . Since $V$ is real, bounded and compactly supported, so is $W$ , and in particular $W \in L^{2} (R^{3})$ . Thus [F2] applies to the actual reduced potential $V_{0} = W$ and this already self-adjoint $A$ ; Cook existence is not used as a proof of the self-adjointness it presupposes.

1.2F1F2algebra

For $ψ \in S$ put $τ (t) = ℏ t / (2 m)$ . By [F1], $V e^{- i t H^{f r e e} / ℏ} ψ = c W e^{- i τ (t) A_{0}} ψ$ , and by [F2] the function $τ \mapsto ∥ W e^{- i τ A_{0}} ψ ∥_{2}$ is integrable on $[0, \infty)$ ; the substitution $t \mapsto τ (t)$ in the Lebesgue integral of a continuous integrand, In one dimension the compact-Jordan formula is substitution over the unoriented image interval with the absolute derivative, gives $\int_{0}^{\infty} ∥ W e^{- i τ (t) A_{0}} ψ ∥_{2} d t = \frac{2 m}{ℏ} \int_{0}^{\infty} ∥ W e^{- i τ A_{0}} ψ ∥_{2} d τ < \infty$ , hence the first Cook integral of (a); the negative half-line is identical.

1.3F2F1given

By [F1] the strong limits in physical time exist exactly when the energy-divided limits in [F2] exist and then coincide; [F2] exhibits them, so $Ω_{\pm}$ of the statement exist as strong limits and are the operators of [F2]. This is (b)'s existence assertion.

1.4F1F4algebra

Let $ψ \in D (H)$ with $H ψ = λ ψ$ and $λ > 0$ . By [F1] $ψ$ is an eigenvector of $A$ with eigenvalue $λ / c > 0$ ; [F4] applied with $k = \sqrt{λ / c} > 0$ gives $ψ = 0$ . This is (d).

2.1F3step 1.3algebra

By [F3] the operators of step 1.3 satisfy $Ω_{\pm}^{*} Ω_{\pm} = I$ , $e^{- i s H / ℏ} Ω_{\pm} = Ω_{\pm} e^{- i s H^{f r e e} / ℏ}$ (after replacing the energy-divided time by the physical time through the same reparametrisation) and $H Ω_{\pm} ψ = Ω_{\pm} H^{f r e e} ψ$ for $ψ \in D (H^{f r e e})$ . This completes (b).

3.1F1F2step 1.4algebra∎

By [F2] the scalar measures $μ_{ψ}^{A_{0}}$ are absolutely continuous with respect to Lebesgue measure with zero mass at $0$ ; by [F1] $μ_{ψ}^{H^{f r e e}} (B) = μ_{ψ}^{A_{0}} (c^{- 1} B)$ , and since multiplication by $c^{- 1} > 0$ preserves Lebesgue null sets, the transfer preserves absolute continuity and atomlessness, and $μ_{ψ}^{H^{f r e e}} ({0}) = μ_{ψ}^{A_{0}} ({0}) = 0$ ; the spectrum is contained in $[0, \infty)$ because $H^{f r e e} = c A_{0}$ with $A_{0} \geq 0$ . Hence every vector is absolutely continuous, $H_{a c} (H^{f r e e}) = H_{0}$ , and there are no eigenvectors. This is (c) and completes the proof.

Remarks

What is not claimed. Ranges, asymptotic completeness and unitarity of $S$ are deliberately absent; they are the content of Asymptotic completeness and the unitary scattering operator for real bounded compactly supported potentials and Asymptotic completeness for the resolvent-tail short-range class, with the decaying-tail example and the explicit exclusion of the Coulomb tail. The Cook estimates are stated for Schwartz vectors exactly as in the cited mathematics; density then gives the strong limits on all of $H_{0}$ .
Units. The displayed Cook integrals carry the dimensions of the integrand times time; no numerical value of $m$ , $ℏ$ or $V$ is asserted.

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Asymptotic completeness and the unitary scattering operator for real bounded compactly supported potentials

Physical scope

The declared short-range scattering model with a real bounded compactly supported potential: asymptotic completeness of the model's wave operators, the vanishing of the singular continuous spectral subspace, the identification of the two ranges with the absolutely continuous and continuous subspaces of the interacting operator, and unitarity of the model's scattering operator with its commutation with the free flow. Conditional on P1, P2, P4, P5, P6, P7, the declared model and the cited mathematics; the item asserts no cross section, no preparation, no detector and no empirical claim.

Statement

Adopt the declared short-range scattering model The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units with mass $m > 0$ , spin zero, action constant $ℏ > 0$ and a declared compact-support-class potential $V$ supported in a ball $B_{R}$ . Let $Ω_{\pm}$ and $S$ be the wave operators and scattering operator of clause (d) of that definition, $E_{H}$ the projection valued measure of the self-adjoint interacting operator $H$ , and $μ_{ψ}^{H}$ its scalar measures. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Then:

(a) the wave operators exist by Existence of the model's wave operators, isometry and intertwining, and the purely absolutely continuous free spectrum, and $ran Ω_{\pm} \subseteq H_{a c} (H)$ and $H_{s c} (H) = {0}$ , where $H_{s c} (H) : = {ψ \in H_{0} : μ_{ψ}^{H} is atomless and singular with respect to Lebesgue measure};$

(b) each wave operator is asymptotically complete, $ran Ω_{+} = ran Ω_{-} = H_{a c} (H) = H_{c} (H);$

(c) $S = Ω_{+}^{*} Ω_{-}$ is unitary on $H_{0}$ and commutes with the free flow, $S e^{- i t H^{f r e e} / ℏ} = e^{- i t H^{f r e e} / ℏ} S$ for every $t \in R$ .

The statement is a conditional consequence of the declared model with a compactly supported real bounded potential; it asserts no stationary amplitude, no partial-wave decomposition, no cross section and no measured quantity.

Facts & Assumptions

Given: the declared short-range scattering model with $m, ℏ > 0$ , spin zero and a compact-support-class potential $V$ supported in $B_{R}$ ; the choice hypotheses.

[F1]

Scaling correspondence in physical coordinates. With $c = ℏ^{2} / (2 m)$ and $W : = 2 m V / ℏ^{2}$ , the operators $A = - Δ_{w} + M_{W}$ and $H$ on $W^{2, 2}$ satisfy $H = c A$ , $E_{H} (c B) = E_{A} (B)$ for every Borel $B$ , the strong wave-operator limits in physical and energy-divided time coincide, and the sets $H_{a c}$ , $H_{s c}$ , $H_{p p}$ , $H_{c}$ of $H$ and of $A$ are equal (Unit, time and spectral correspondence between the physical scattering operators and the nondimensional Laplacian clauses (a), (b), (d), (e)).

[F2]

Compact-support asymptotic completeness. Assume Countable Choice. For real bounded $V_{0}$ with support in a ball, the energy-divided operator $- Δ + M_{V_{0}}$ on $H^{2} (R^{3})$ has wave operators with $ran Ω_{\pm} \subseteq H_{a c}$ , $H_{s c} = {0}$ , $ran Ω_{+} = ran Ω_{-} = H_{a c} = H_{c}$ , and $S = Ω_{+}^{*} Ω_{-}$ unitary and commuting with the free flow (Compact-support asymptotic completeness clauses (a), (b), (c)).

[F3]

Scattering-state vocabulary. The model reads its wave operators and $S$ exactly as the strong limits and the product of clause (d) of its definition, and reads asymptotic completeness as the equality of the ranges with $H_{a c} (H)$ and $H_{c} (H)$ (The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units clause (d)).

Proof

technique · direct; all coordinates in this proof are physical $x$, with $W=W_x$ and the auxiliary parameter $tc/\hbar$ in square metres; the abstract mathematical supplier is applied to their numerical values in the fixed metre convention; apply the energy-divided compact-support completeness theorem to the rescaled potential and transfer the range, spectral-subspace and unitarity statements to the model through the scaling correspondence

Given: the data and hypotheses of the statement and [F1]-[F3].

1.1F1F2given

The declared potential $V$ is real, bounded and supported in $B_{R}$ , so $W = V / c$ is real, bounded and supported in the same ball; hence [F2] applies to $W$ . By [F1] the model's interacting operator is $H = c A$ with $A = - Δ_{w} + M_{W}$ on the common domain.

1.2F1F2given

By [F2] the energy-divided wave operators exist with ranges inside $H_{a c} (A)$ and with $H_{s c} (A) = {0}$ ; by [F1] the physical strong limits exist exactly together with the energy-divided ones and coincide, and the spectral subspaces of $H$ and $A$ agree; this gives (a).

2.1F2step 1.2F1algebra

By [F2], $ran Ω_{+} = ran Ω_{-} = H_{a c} (A) = H_{c} (A)$ for the energy-divided operators; by [F1] the same displayed equalities hold with $H$ in place of $A$ and the same operators, which is (b).

3.1F2step 2.1F1F3algebra∎

By [F2] the operator $Ω_{+}^{*} Ω_{-}$ for the energy-divided pair is unitary and commutes with the free flow; since the physical and energy-divided wave operators coincide by [F1], $S = Ω_{+}^{*} Ω_{-}$ is unitary and commutes with $e^{- i t H^{f r e e} / ℏ}$ for every $t$ . This is (c) and completes the proof.

Remarks

No uniform threshold statement. The cited mathematics includes a threshold obstruction: for compactly supported potentials the compactness of the scattering defects fails uniformly at zero energy unless the energy cutoff vanishes there (Compact scattering defects and the zero-energy obstruction). Completeness itself, as stated here, is not affected; the obstruction concerns the uniform compact-defect statement, not the range equalities.
Zero-energy scattering. The theorem asserts nothing about the behaviour of the stationary solutions at $k = 0$ ; every stationary statement of this page is made for $k > 0$ .

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Asymptotic completeness for the resolvent-tail short-range class, with the decaying-tail example and the explicit exclusion of the Coulomb tail

Physical scope

The declared short-range scattering model with a potential in the resolvent-tail class (square-integrable plus bounded decomposition whose shifted nondimensional free resolvent tail is integrable): self-adjointness, existence and asymptotic completeness of the model's wave operators, vanishing singular continuous spectrum, and unitarity of the scattering operator; together with the explicitly checked sufficient decay class |V_tail| <= C(1+|x|/L*)^(-1-delta), delta>0, with reference length L*>0 and C in joules, and the explicit exclusion of the Coulomb tail delta=0 from this class. Conditional on the declared model and the cited mathematics; no cross section, preparation, detector or empirical claim.

Statement

Adopt the declared short-range scattering model The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units with mass $m > 0$ , spin zero, action constant $ℏ > 0$ , and a declared potential $V$ of the resolvent-tail class: real $V = V_{2} + V_{b}$ with $V_{2} \in L^{2} (R^{3})$ and $V_{b} \in L^{\infty} (R^{3})$ real, and, with the model's reference length $L_{*}$ , energy $E_{*} = ℏ^{2} / (2 m L_{*}^{2})$ , dimensionless coordinate $y = x / L_{*}$ , reduced potential $W_{*} (y) = V (L_{*} y) / E_{*}$ and the nondimensional free operator $A_{0, *} : = - Δ_{y}^{w e a k}$ of the model, $h_{*} (r) : = ∥ M_{W_{*}} (A_{0, *} - i I)^{- 1} 1_{{∣ y ∣ \geq r}} ∥,$ finite for every dimensionless $r \geq 0$ and in $L^{1} ([0, \infty))$ . Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Then:

(a) $H = H^{f r e e} + M_{V}$ is self-adjoint on $D (H) = W^{2, 2} (R^{3}; C)$ (Integer-order Sobolev spaces and their norms); both wave operators $Ω_{\pm} = {s-lim}_{t \to \pm \infty} e^{i t H / ℏ} e^{- i t H^{f r e e} / ℏ}$ exist as bounded linear isometries, $ran Ω_{+} = ran Ω_{-} = H_{a c} (H) = H_{c} (H)$ , and $H_{s c} (H) = {0}$ ;

(b) $S = Ω_{+}^{*} Ω_{-}$ is unitary on $H_{0}$ ;

(c) the conclusions of (a) and (b) hold in particular for a declared potential of the explicit decaying-tail form $V = V_{l o c} + V_{t a i l}$ with real $V_{l o c} \in L^{2} (R^{3})$ compactly supported and $∣ V_{t a i l} (x) ∣ \leq C (1 + ∣ x ∣ / L_{*})^{- 1 - δ}$ almost everywhere for some constants $C > 0$ in joules and dimensionless $δ > 0$ ; here the integrability hypothesis of the preamble is not presupposed, because the cited theorem supplies it for this form;

(d) the explicit decaying-tail form of (c) does not cover the pure Coulomb potential $V (x) = - κ / ∣ x ∣$ , $κ > 0$ : after any split into a compactly supported $L^{2}$ part and a tail part, the bound $∣ V_{t a i l} (x) ∣ \leq C (1 + ∣ x ∣ / L_{*})^{- 1 - δ}$ fails for every $δ > 0$ , because $κ (1 + ∣ x ∣ / L_{*})^{1 + δ} / ∣ x ∣ \to + \infty$ as $∣ x ∣ \to \infty$ . The pure Coulomb potential also fails the general resolvent-tail integrability hypothesis itself, as proved below; the Coulomb case is the separate declared long-range model of this page.

The statement is conditional on the declared model with the displayed integrability hypothesis and the cited mathematics; no cross section, stationary amplitude or measured quantity is asserted.

Facts & Assumptions

Given: the declared short-range scattering model with $m, ℏ > 0$ , spin zero and a resolvent-tail-class potential $V$ ; the choice hypotheses.

[F1]

Spatial and energy scaling. The model proves that the unitary $D_{*}$ maps $W_{x}^{2, 2}$ onto $W_{y}^{2, 2}$ and conjugates the physical pair to $E_{*} A_{0, *}$ and $E_{*} A_{*}$ , where $A_{*} = A_{0, *} + M_{W_{*}}$ . The purely mathematical positive-scaling correspondence Unit, time and spectral correspondence between the physical scattering operators and the nondimensional Laplacian applies to these operators with positive scalar $c = E_{*}$ (its numerical parameters can be chosen as $ℏ^{'} = ℏ$ , $m^{'} = ℏ^{2} / (2 E_{*})$ ). It gives spectral dilation, preservation of spectral types and the time change $τ = t E_{*} / ℏ = t / T_{*}$ . Unitary conjugation transports the PVM and the strong limits, so $E_{H} (E_{*} B) = D_{*}^{*} E_{A_{*}} (B) D_{*}$ and $Ω_{\pm} = D_{*}^{*} Ω_{\pm, *} D_{*}$ ; each physical spectral subspace is $D_{*}^{*}$ of the corresponding subspace for $A_{*}$ .

[F2]

Resolvent-tail short-range completeness in dimensionless coordinates. Assume Countable Choice. For real $V_{0} = V_{0, 2} + V_{0, b}$ with $V_{0, 2} \in L^{2}$ , $V_{0, b} \in L^{\infty}$ real and $∥ M_{V_{0}} (- Δ - i)^{- 1} 1_{{∣ x ∣ \geq r}} ∥$ finite for every dimensionless $r \geq 0$ and integrable over $r \geq 0$ , the operator $- Δ + M_{V_{0}}$ on $H^{2} (R^{3}; C)$ is self-adjoint, both ordinary wave operators exist, $H_{s c} = {0}$ , $ran Ω_{+} = ran Ω_{-} = H_{a c} = H_{c}$ , and $S = Ω_{+}^{*} Ω_{-}$ is unitary; all conclusions hold for $V_{0} = V_{0, l o c} + V_{0, t a i l}$ with compactly supported $L^{2}$ part and $∣ V_{0, t a i l} (x) ∣ \leq C (1 + ∣ x ∣)^{- 1 - δ}$ , $δ > 0$ (Resolvent-tail short-range completeness clause display and its final paragraph).

[F3]

The model's readings. The model's wave operators and scattering operator are the strong limits and the product of clause (d) of its definition, with the potential classes declared in clause (b) (The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units clauses (b), (d)).

[F4]

A nonzero smooth bump supported in a fixed ball exists (A smooth bump between concentric Euclidean balls). Translation preserves complex $L^{2}$ norms (Complex translation, convolution, approximate identities, and mollification, first clause); on compact smooth functions the free operator is exactly $- Δ$ on $H^{2}$ (The three-dimensional free Schrödinger operator as the weak Laplacian on its Sobolev domain (a) at $m = 1 / 2$ , $ℏ = 1$ ). Coordinate differentiation commutes with translation by direct substitution in the derivative definition.

Proof

technique · direct; the class hypothesis is exactly the nondimensional resolvent-tail hypothesis for the reduced potential, so the cited theorem applies, and the scaling correspondence transfers self-adjointness, the limits, the range equalities and the unitarity

Given: the data and hypotheses of the statement and [F1]-[F4].

1.1F1F2givenalgebra

Put $W_{*} (y) = V (L_{*} y) / E_{*} = W_{*, 2} (y) + W_{*, b} (y)$ . Lebesgue dilation gives $∥ W_{*, 2} ∥_{2} = E_{*}^{- 1} L_{*}^{- 3 / 2} ∥ V_{2} ∥_{2}$ and $∥ W_{*, b} ∥_{\infty} = E_{*}^{- 1} ∥ V_{b} ∥_{\infty}$ . Thus these parts are real $L^{2}$ and bounded functions respectively. In [F2] use the dimensionless variable $y$ in place of its abstract variable $x$ , $V_{0} = W_{*}$ and $A_{0, *} = - Δ_{y}^{w e a k}$ . The full hypothesis is exactly $h_{*} (r) < \infty$ for every $r$ with $h_{*} \in L^{1}$ , so [F2] gives self-adjoint $A_{*}$ on $H_{y}^{2}$ , isometric wave operators $Ω_{\pm, *}$ , $H_{s c} (A_{*}) = 0$ , equal ranges $H_{a c} (A_{*}) = H_{c} (A_{*})$ and unitary $S_{*}$ .

1.2F4givenalgebra

[Failure of the general Coulomb resolvent-tail condition] For the pure Coulomb reduced potential put $W_{*, C} = - γ_{*} / ∣ y ∣$ , $γ_{*} = κ / (E_{*} L_{*}) > 0$ . All radii in this argument are dimensionless. Fix by [F4] a nonzero real $ϕ \in C_{c}^{\infty} (B_{R})$ , and for $r > R$ set $ϕ_{r} (y) = ϕ (y - 2 r e_{1})$ and $f_{r} = (A_{0, *} - i) ϕ_{r}$ . Both are supported outside the radius- $r$ ball, since $2 r - R > r$ ; derivatives do not enlarge support. The free derivative action and translation isometry give $∥ ϕ_{r} ∥_{2} = ∥ ϕ ∥_{2} > 0$ and $∥ f_{r} ∥_{2} = ∥ (A_{0, *} - i) ϕ ∥_{2} = : c_{f} > 0$ . Positivity and self-adjointness of $A_{0, *}$ ensure its resolvent at $i$ exists, and $(A_{0, *} - i)^{- 1} 1_{{∣ y ∣ \geq r}} f_{r} = ϕ_{r}$ . If the Coulomb resolvent-tail operator is not bounded at some radius it already fails the class; otherwise its operator norm obeys $h_{*, C} (r) \geq \frac{∥ M_{W_{*, C}} ϕ_{r} ∥_{2}}{∥ f_{r} ∥_{2}} \geq \frac{γ_{*} ∥ ϕ ∥_{2}}{c_{f} (2 r + R)} .$ The last bound uses $∣ y ∣ \leq 2 r + R$ on that support. The positive lower bound has divergent integral over $r > R$ , so $h_{*, C} \notin L^{1} ([0, \infty))$ . Thus pure Coulomb fails the general class as well, proving the general-class exclusion in (d) without any zero-energy hypothesis.

2.1F1step 1.1algebra

By [F1], $H = D_{*}^{*} E_{*} A_{*} D_{*}$ is self-adjoint on $W_{x}^{2, 2}$ . For $τ = t / T_{*}$ the physical comparison is $D_{*}^{*} e^{i τ A_{*}} e^{- i τ A_{0, *}} D_{*}$ . Since $T_{*} > 0$ , each physical infinite-time limit exists and equals $D_{*}^{*} Ω_{\pm, *} D_{*}$ . These are isometries, their ranges are the conjugated spectral subspaces $H_{a c} (H) = H_{c} (H)$ and $H_{s c} (H) = 0$ . This proves (a).

3.1F1F3step 1.1step 2.1algebra

By [F3] and step 2.1, $S = D_{*}^{*} S_{*} D_{*}$ , which is unitary since $S_{*}$ and $D_{*}$ are unitary. This proves (b).

4.1F1F2step 2.1step 3.1algebra

For the physical decay form in (c), set $W_{*, l o c} (y) = V_{l o c} (L_{*} y) / E_{*}$ and $W_{*, t a i l} (y) = V_{t a i l} (L_{*} y) / E_{*}$ . The local part has norm $E_{*}^{- 1} L_{*}^{- 3 / 2} ∥ V_{l o c} ∥_{2}$ and support radius divided by $L_{*}$ . The tail satisfies $∣ W_{*, t a i l} (y) ∣ \leq (C / E_{*}) (1 + ∣ y ∣)^{- 1 - δ}$ . Thus [F2]'s full sufficient-decay result supplies the integrable resolvent-input tail and all conclusions for $A_{*}$ ; step 2.1 and step 3.1 transfer them to the physical operators. This proves (c) without assuming the resolvent estimate for this example.

5.1F2step 1.2step 4.1givenalgebra∎

For $V (x) = - κ / ∣ x ∣$ with $κ > 0$ and any split $V = V_{l o c} + V_{t a i l}$ into a compactly supported part and the remainder, the remainder equals $- κ / ∣ x ∣$ for all sufficiently large $∣ x ∣$ , so for every $δ > 0$ the quotient $(1 + ∣ x ∣ / L_{*})^{1 + δ} ∣ V_{t a i l} (x) ∣ = κ (1 + ∣ x ∣ / L_{*})^{1 + δ} / ∣ x ∣$ tends to $+ \infty$ as $∣ x ∣ \to \infty$ ; hence no constant $C$ satisfies the sufficient decay bound of [F2]'s final paragraph, and the explicit form of (c) does not cover the Coulomb potential. This proves the sufficient-decay exclusion in (d).

Remarks

What the class excludes. The resolvent-tail class excludes Coulomb's $δ = 0$ tail; the long-range comparison construction for the Coulomb potential is a separate declared model of this page (Completeness of the Dollard-modified wave operators for the declared Coulomb model).
Threshold. No zero-energy nonresonance or absence-of-zero-eigenvalue hypothesis is imposed by the cited theorem; the item makes no statement about the stationary problem at $k = 0$ .

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The model's stationary scattering wave, its outgoing amplitude and the differential and total cross-section readings

Physical scope

The declared short-range scattering model with a real bounded compactly supported potential, in the pair's stationary elastic beam reading at fixed wave number and incident direction: existence and uniqueness of the outgoing stationary solution on the interaction region, its local regularity, its uniform far-field expansion with a smooth amplitude and O(r^-2) remainder, the scattered radial current, and the resulting differential and total cross-section readings in m^2 per steradian and m^2. Conditional on the declared model and the cited mathematics; the stationary wave is not a state-space element and the item asserts no preparation, no detector acceptance, no count rate and no measured cross section.

Statement

Adopt the declared short-range scattering model The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units with mass $m > 0$ , spin zero and action constant $ℏ > 0$ , with a compact-support-class potential $V$ supported in the ball $B_{R}$ , and adopt the declared stationary elastic beam model The declared stationary elastic beam model: incident plane wave, scattered wave, current readings and differential and total cross sections at wave number $k > 0$ , direction $d \in S^{2}$ and the declared $n_{i n c} > 0$ in $m^{- 3}$ , with incident energy $E = ℏ^{2} k^{2} / (2 m)$ . The construction below retains the dimensionless unit-incident shape $Ψ$ ; its physical beam field is $Φ = \sqrt{n_{i n c}} Ψ$ . Put $W : = \frac{2 m}{ℏ^{2}} V, G_{k} (z) : = \frac{e^{i k ∣ z ∣}}{4 π ∣ z ∣}, (K_{k} f) (x) : = \int_{B_{R}} G_{k} (x - y) W (y) f (y) d y$ for $f \in L^{2} (B_{R}; C)$ and $x \in B_{R}$ . Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Then:

(a) $I + K_{k}$ is boundedly invertible on $L^{2} (B_{R}; C)$ ; with $u : = (I + K_{k})^{- 1} (e^{i k d \cdot x} ∣_{B_{R}})$ and $Ψ (x) : = e^{i k d \cdot x} - \int_{B_{R}} G_{k} (x - y) W (y) u (y) d y,$ one has $Ψ \in H_{l o c}^{2} (R^{3})$ , $(- Δ + W - k^{2}) Ψ = 0$ distributionally on $R^{3}$ , and $Ψ ∣_{B_{R}} = u$ ;

(b) with $r = ∣ x ∣ > 2 R$ and $Ω = x / r$ , the scattering amplitude $f_{k} (Ω) : = - \frac{1}{4 π} \int_{B_{R}} e^{- i k Ω \cdot y} W (y) u (y) d y$ is smooth on $S^{2}$ and $Ψ (x) = e^{i k d \cdot x} + \frac{e^{i k r} f_{k} (Ω)}{r} + E (x),$ where $∣ E (x) ∣ + ∣ \partial_{r} E (x) ∣ \leq C (k, R, ∥ W u ∥_{1}) r^{- 2}$ uniformly in $Ω$ , and for every fixed product $D_{Ω}$ of smooth tangential differential operators on $S^{2}$ also $∣ D_{Ω} E ∣ + ∣ D_{Ω} \partial_{r} E ∣ \leq C_{D} r^{- 2}$ ;

(c) the physical scattered radial current of the beam model, with $Φ_{s c} = \sqrt{n_{i n c}} (Ψ - e^{i k d \cdot x})$ , is $j_{r}^{s c} (r, Ω) = n_{i n c} (\frac{ℏ k}{m} \frac{∣ f_{k} (Ω) ∣^{2}}{r^{2}} + O (r^{- 3}))$ uniformly in $Ω$ , with physical incident flux density $J_{i n c} = n_{i n c} ℏ k / m$ in $m^{- 2} s^{- 1}$ ; the common density factor cancels, and the cross-section readings are $\frac{d σ}{d Ω} (Ω) = ∣ f_{k} (Ω) ∣^{2} in m^{2} s r^{- 1}, σ = \int_{S^{2}} ∣ f_{k} (Ω) ∣^{2} d Ω < \infty in m^{2};$

(d) the solution $Ψ$ of (a) is a generalized stationary solution of the beam model at energy $E$ and direction $d$ in the sense of clause (c) of The declared stationary elastic beam model: incident plane wave, scattered wave, current readings and differential and total cross sections, and no other function satisfying the same equation on the complement of $supp V$ and the same outgoing reading is produced by the construction.

Every assertion is conditional on the declared model and the cited mathematics. The stationary wave is not an element of the state space; the item asserts no preparation, no beam apparatus, no detector acceptance and no measured count rate.

Facts & Assumptions

Given: the declared model and beam model with $m, ℏ > 0$ , compact-support-class $V$ supported in $B_{R}$ , wave number $k > 0$ , direction $d$ , density scale $n_{i n c} > 0$ , reduced potential $W = (2 m / ℏ^{2}) V$ , kernel $G_{k}$ , operator $K_{k}$ and choice hypotheses.

[F1]

Stationary outgoing integral and flux. Assume Countable Choice. Let $R > 0$ , real $V_{0} \in L^{\infty}$ with support in $B_{R}$ and $k > 0$ ; with $K_{k}$ built from $V_{0}$ and $G_{k}$ as above (with $V_{0}$ in place of $W$ ), the operator $I + K_{k}$ is boundedly invertible, $u = (I + K_{k})^{- 1} e_{i n c}$ , $Ψ = e_{i n c} - \int G_{k} (\cdot - y) V_{0} (y) u (y) d y$ satisfies $Ψ \in H_{l o c}^{2}$ , $(- Δ + V_{0} - k^{2}) Ψ = 0$ distributionally and $Ψ ∣_{B_{R}} = u$ ; the amplitude $f_{k} (Ω) = - (4 π)^{- 1} \int e^{- i k Ω \cdot y} V_{0} (y) u (y) d y$ is smooth with the displayed uniform far-field and derivative bounds, and the scattered radial current and cross-section displays hold (Stationary outgoing integral and flux clauses (a), (b), (c)).

[F2]

Fredholm invertibility. Assume Countable Choice. For real bounded compactly supported $V_{0}$ and every $k > 0$ , the operator $I + K_{k}$ of [F1] is boundedly invertible (Outgoing Fredholm invertibility).

[F3]

Scaling correspondence. With $c = ℏ^{2} / (2 m)$ , the model's energy operator satisfies $H = c A$ , $A = - Δ_{w} + M_{W}$ with $W = 2 m V / ℏ^{2}$ ; the physical energy $E$ and the energy-divided spectral value $k^{2}$ in $m^{- 2}$ correspond by $E = c k^{2}$ , i.e. $E = ℏ^{2} k^{2} / (2 m)$ (Unit, time and spectral correspondence between the physical scattering operators and the nondimensional Laplacian clauses (a), (b)).

[F4]

Unit-shape and physical current readings. The mathematical current functional is $j^{u n i t} [ψ] = (ℏ / m) Im (\overline{ψ} \nabla ψ)$ . For the unit incident shape its value is $(ℏ k / m) d$ , and for the exact outgoing shape $f e^{i k r} / r$ its radial value is $(ℏ k / m) ∣ f ∣^{2} / r^{2}$ (The three-dimensional continuity identity, the vanishing stationary divergence and the incident and outgoing current readings clause (c)). For the declared constant $n_{i n c} > 0$ , the beam model reads the physical field as $\sqrt{n_{i n c}} ψ$ , so the quadratic functional gives $j [\sqrt{n_{i n c}} ψ] = n_{i n c} j^{u n i t} [ψ]$ , and its cross sections are ratios with $J_{i n c} = n_{i n c} ℏ k / m$ (The declared stationary elastic beam model: incident plane wave, scattered wave, current readings and differential and total cross sections (b)-(d)).

[F5]

The beam model's solubility vocabulary. A generalized stationary solution belongs to $H_{l o c}^{2} (R^{3})$ , has $V Ψ \in L_{l o c}^{2}$ , satisfies the stationary equation distributionally on $R^{3}$ , and is $C^{2}$ on the complement of the essential closed support of $V$ . On this exterior it satisfies the free equation $- ℏ^{2} Δ Ψ / (2 m) = E Ψ$ pointwise; the equation with the declared $V$ holds almost everywhere, or pointwise after replacing $V$ by its zero exterior representative. Its outgoing reading has the specified amplitude and remainder bounds (The declared stationary elastic beam model: incident plane wave, scattered wave, current readings and differential and total cross sections clause (c)).

Proof

technique · direct; the physical stationary equation divided by $c=\hbar^{2}/(2m)$ is the energy-divided equation in physical coordinates with potential $W=W_x$, so the cited mathematical stationary theorem applies verbatim and its amplitude and flux displays are exactly the model's readings

Given: the data and hypotheses of the statement and [F1]-[F5].

1.1F3givenalgebra

Since $V$ is real, bounded and supported in $B_{R}$ , so is $W = 2 m V / ℏ^{2}$ , which is read in $m^{- 2}$ ; dividing the physical stationary equation $- \frac{ℏ^{2}}{2 m} Δ Ψ + V Ψ = E Ψ$ by $c = ℏ^{2} / (2 m)$ gives $- Δ Ψ + W Ψ = (E / c) Ψ$ , and $E / c = ℏ^{2} k^{2} / (2 m) \cdot 2 m / ℏ^{2} = k^{2}$ by [F3] with $E = ℏ^{2} k^{2} / (2 m)$ .

2.1F2F1step 1.1algebra

The kernel $G_{k} (x - y) W (y)$ on $B_{R} \times B_{R}$ is square-integrable because $∣ x - y ∣^{- 2}$ is locally integrable in three dimensions and $B_{R}$ is bounded, so $K_{k}$ is Hilbert-Schmidt and [F2] gives bounded invertibility of $I + K_{k}$ ; [F1] then gives $Ψ \in H_{l o c}^{2}$ , the distributional equation $(- Δ + W - k^{2}) Ψ = 0$ and $Ψ ∣_{B_{R}} = u$ , which is (a).

3.1F1step 2.1

By [F1] the smooth amplitude $f_{k} (Ω) = - (4 π)^{- 1} \int e^{- i k Ω \cdot y} W (y) u (y) d y$ and the uniform expansion with the $r^{- 2}$ remainder and its tangential and radial derivative bounds hold; these are exactly the displays of (b), with the constant depending on $k, R$ and $∥ W u ∥_{1}$ .

4.1F1F4step 3.1algebra

By [F1] the unit-shape scattered field $Ψ_{s c} = Ψ - e^{i k d \cdot x}$ has radial mathematical current $(ℏ k / m) ∣ f_{k} ∣^{2} / r^{2} + O (r^{- 3})$ uniformly. Write $Ψ_{s c} = u + i v$ ; the physical field has real and imaginary parts $\sqrt{n_{i n c}} u, \sqrt{n_{i n c}} v$ . Since the scale is constant, its current is $(ℏ / m) [(\sqrt{n_{i n c}} u) \nabla (\sqrt{n_{i n c}} v) - (\sqrt{n_{i n c}} v) \nabla (\sqrt{n_{i n c}} u)] = n_{i n c} (ℏ / m) (u \nabla v - v \nabla u)$ by [F4]. Thus the physical scattered current is the display in (c), and the same calculation on the unit incident plane wave gives $J_{i n c} = n_{i n c} ℏ k / m > 0$ . Dividing yields $\lim_{r \to \infty} r^{2} j_{r}^{s c} / J_{i n c} = ∣ f_{k} ∣^{2}$ . Smoothness and boundedness of $f_{k}$ on the sphere, with [F1]'s uniform error, give the finite total cross section $σ = \int_{S^{2}} ∣ f_{k} ∣^{2} d Ω$ . The unit-shape amplitude stays in metres and the density cancels from both cross-section readings. This is (c).

5.1F3F5step 2.1step 3.1algebra∎

By step 2.1 the constructed $Ψ$ lies in $H_{l o c}^{2}$ and satisfies the global distributional stationary equation after the scaling of [F3]; boundedness of $V$ gives $V Ψ \in L_{l o c}^{2}$ . On the complement of its essential closed support, the convolution integrates the compactly supported $L^{1}$ density $W u$ at a positive distance from the kernel singularity locally, so differentiation under the integral gives a smooth field. Here $V = 0$ almost everywhere; thus the continuous free-equation residual vanishes distributionally and hence pointwise, as in [F5]. The equation with the declared representative $V$ therefore holds almost everywhere there, and pointwise for its zero exterior representative. Step 3.1 gives the outgoing reading and all required remainder bounds. These facts verify the beam model's generalized stationary solution conditions. The invertible integral problem uniquely fixes the constructed $u$ and hence $Ψ$ , which proves the construction-specific uniqueness in (d) and completes the proof.

Remarks

What is read, and what is not. The amplitude and cross sections are computed from the explicitly constructed stationary field; the pair asserts nothing about a preparation that would realize the incident plane-wave reading, and no measured count rate is predicted. Converting these model quantities into an expected count rate requires a declared target, fluence and detector model with its own uncertainties.
Normalization. With physical coordinates $x$ in metres and $k$ in inverse metres, the unit-incident shape $Ψ$ is dimensionless, the reduced potential $W = (2 m / ℏ^{2}) V$ is in $m^{- 2}$ , and the displayed integral for $f_{k}$ has dimension length. The physical beam field $\sqrt{n_{i n c}} Ψ$ is in $m^{- 3 / 2}$ ; its number currents are in $m^{- 2} s^{- 1}$ . The common density factor cancels, leaving $d σ / d Ω$ in $m^{2} s r^{- 1}$ independently of its value.

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The model's partial-wave and phase-shift vocabulary for a declared central potential

Physical scope

The pair's declared partial-wave and phase-shift vocabulary for a declared central short-range potential: the spherical harmonic channel labels, the normalized regular and outgoing radial solutions, the unit-modulus channel factor, the phase shift read modulo pi, and the channel cross-section reading. The definition is a reading of the stationary model of the pair; it asserts no angular-momentum interpretation of the labels for a material system, no bound-state statement and no measured quantity.

Definition

(a) Framework data. Adopt the declared short-range scattering model The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units with mass $m > 0$ , spin zero, action constant $ℏ > 0$ , and its stationary elastic beam model The declared stationary elastic beam model: incident plane wave, scattered wave, current readings and differential and total cross sections at wave number $k > 0$ ; adopt the declared central-potential reading of The declared central-potential model: state space, assigned energy operator, spherical reading and bound-state vocabulary: a declared central potential of the pair is a real bounded measurable $V (x) = v (∣ x ∣)$ with $v : (0, \infty) \to R$ bounded, and it is short-range when it lies in the compact-support class of the scattering model. Units are those of Dimensions, SI unit conventions and quantity types of the framework; the wave number $k$ is read in $m^{- 1}$ and the energy $E = ℏ^{2} k^{2} / (2 m)$ in joules.

(b) Channel vocabulary. Let $S^{2} = {ω \in R^{3} : ∣ ω ∣ = 1}$ be the sphere of spatial directions, with polar surface measure $σ$ and angle coordinates $Φ (θ, α) = (\sin θ \cos α, \sin θ \sin α, \cos θ)$ ; $d σ = \sin θ d θ d α$ and $σ (S^{2}) = 4 π$ . Use exactly the dimensionless spherical harmonics defined in Associated Legendre function normalization: for integers $l \geq 0$ and $0 \leq q \leq l$ , set $P_{l} (t) = \frac{1}{2^{l} l!} \frac{d^{l}}{d t^{l}} (t^{2} - 1)^{l}, w_{l q} (t) = (- 1)^{q} (1 - t^{2})^{q / 2} \frac{d^{q}}{d t^{q}} P_{l} (t), b_{l q} = {(\int_{- 1}^{1} w_{l q} (t)^{2} d t)}^{- 1 / 2} > 0.$ The continuous representatives $Y_{l q} : S^{2} \to C$ are given by $Y_{l q} (Φ (θ, α)) = \frac{b_{l q}}{\sqrt{2 π}} w_{l q} (\cos θ) e^{i q α} (0 < θ < π, 0 \leq α < 2 π),$ extended continuously to the poles; for $1 \leq q \leq l$ set $Y_{l, - q} = (- 1)^{q} \overline{Y_{l q}}$ . The same symbols denote their almost-everywhere classes in $L^{2} (S^{2}, σ; C)$ . Under the declared Countable Choice assumption, Sphere ONB proves that these classes are orthonormal for $⟨ F, G ⟩ = \int_{S^{2}} F \overline{G} d σ$ and their closed linear span is all of $L^{2} (S^{2}, σ; C)$ . For a declared central short-range potential $V$ , an angular channel $(l, \overset{ˉ}{m})$ , with integers $l \geq 0$ and $- l \leq \overset{ˉ}{m} \leq l$ , means the one-dimensional subspace ${span}_{C} {Y_{l \overset{ˉ}{m}}}$ ; the partial wave $l$ means the degree- $l$ sector $H_{l} : = {span}_{C} {Y_{l q} : - l \leq q \leq l}$ . The angular indices $q, \overset{ˉ}{m}$ are distinct from the positive mass $m$ .

The radial solutions are those constructed in Partial-wave cross-section convergence. The reduced equation below is read in physical coordinates after energy division: the reduced potential is $w : = 2 m v / ℏ^{2}$ of clause (c) of The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units and the energy-divided spectral value is $k^{2}$ , both in $m^{- 2}$ , so that the physical parameters $m, ℏ$ do not enter it explicitly. The model declares the regular radial solution $φ_{l}$ of the reduced equation $- φ_{l}^{''} + (w (r) + \frac{l (l + 1)}{r^{2}}) φ_{l} = k^{2} φ_{l}, φ_{l} (r) / r^{l + 1} ⟶ 1 (r \to 0),$ and the outgoing radial solution $f_{l}$ with $f_{l} (r) = e^{i (k r - l π / 2)} (1 + O (r^{- 1})) (r \to \infty)$ outside the interaction region. Both are maps $(0, \infty) \to C$ whose functions and first derivatives are locally absolutely continuous; the reduced equation holds almost everywhere, equivalently in distributions on $(0, \infty)$ , as in the cited theorem's bounded measurable potential construction. They are not asserted to be elements of the state space.

(c) Channel factor and phase shift. The model declares the channel factor and the phase shift of the partial wave $l$ to be $S_{l} (k) : = \frac{\overline{W (φ_{l}, f_{l})}}{W (φ_{l}, f_{l})}, S_{l} (k) = e^{2 i δ_{l} (k)},$ where $W (φ_{l}, f_{l}) = φ_{l} f_{l}^{'} - φ_{l}^{'} f_{l}$ is the Wronskian and $δ_{l} (k)$ is the phase shift, a real number read modulo $π$ ; the model reads $∣ S_{l} (k) ∣ = 1$ and calls the channel elastic. The declared channel cross-section reading is $σ_{l} (k) : = \frac{4 π}{k^{2}} (2 l + 1) \sin^{2} δ_{l} (k),$ read in $m^{2}$ . The existence of the functions and the identities $∣ S_{l} ∣ = 1$ and $\sum_{l} (2 l + 1) \sin^{2} δ_{l} < \infty$ are proved on this page and are not presupposed here.

(d) Status and scope. Clauses (b)-(c) are declarations of the pair: names for the objects constructed by the cited mathematics and the reading of the phase shift modulo $π$ . The model asserts no identification of the labels $l, \overset{ˉ}{m}$ with a measured angular momentum of a material system, no statement about bound states or their count, and no channel decomposition for non-central potentials. Every numerical statement about $δ_{l}$ is a separate theorem or example of the pair.

Remarks

Modulo $π$ . The phase shift is read modulo $π$ , since $S_{l} = e^{2 i δ_{l}}$ determines $δ_{l}$ only up to an integer multiple of $π$ ; every display of this page that uses $δ_{l}$ is insensitive to that ambiguity, as $\sin^{2}$ is.
Non-central potentials. For a non-central declared potential the pair asserts no diagonal channel factor; the on-shell operator of the following theorem is then a general unitary on $L^{2} (S^{2})$ .

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The declared single-channel resonance model: pole-shaped channel factor and its cross-section reading

Physical scope

The declared single-channel resonance model for one degree-l partial-wave sector of the pair's central scattering reading, including all 2l+1 angular labels with their common scalar factor: the pole-shaped unit-modulus channel factor with resonance energy and width parameters read in joules, the induced phase shift, and the channel cross-section reading in m^2. The model is a declared conditional model of one channel; it makes no assertion that a potential produces this factor, claims no decay law or lifetime identification, and asserts no measured line width or count.

Definition

(a) Framework data. Adopt the declared short-range scattering model The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units with mass $m > 0$ , spin zero and action constant $ℏ > 0$ , and its partial-wave and phase-shift reading The model's partial-wave and phase-shift vocabulary for a declared central potential for a declared central short-range potential. Fix a partial-wave label $l \in {0, 1, 2, \dots}$ in the sense of clause (b) of that reading. Here a single channel means the whole degree- $l$ partial-wave sector: $span {Y_{l μ} : - l \leq μ \leq l, μ \in Z}$ , with all $2 l + 1$ angular labels $(l, μ)$ sharing the same scalar factor $S_{l}$ . The symbol $μ$ is the angular index; $m$ remains the positive mass. This convention agrees with that reading's $σ_{l}$ , which includes all these angular labels. Units are those of Dimensions, SI unit conventions and quantity types of the framework.

(b) The declared single-channel resonance model. As an additional model assumption of the pair, fix real resonance parameters $E_{r}$ and $Γ > 0$ , both read in joules, and declare the channel factor of the label $l$ to be $S_{l} (E) : = \frac{E - E_{r} - i Γ / 2}{E - E_{r} + i Γ / 2} (E > 0),$ read for positive energies only, with the phase shift of the channel read from the relation $S_{l} = e^{2 i δ_{l}}$ modulo $π$ and the channel cross-section reading $σ_{l} (E) : = \frac{4 π}{k (E)^{2}} (2 l + 1) \sin^{2} δ_{l} (E), k (E) : = \frac{\sqrt{2 m E}}{ℏ},$ read in $m^{2}$ . The model calls $E_{r}$ the resonance energy and $Γ$ the width parameter; both are declared model parameters and neither is identified with a measured quantity. The parameter $E_{r}$ may be zero or negative; calling it a resonance energy does not assign a physical cross section at that parameter value unless $E_{r} > 0$ . Auxiliary real-axis algebraic comparisons of the Lorentzian factor do not extend $k (E)$ or $σ_{l} (E)$ beyond the positive-energy domain.

(c) Status and scope. Clauses (a)-(b) declare a conditional one-channel model. The pair asserts no analytic continuation, no pole of the resolvent and no property of any potential that would produce the declared factor $S_{l}$ ; the model is used only to compute the induced phase shift and channel cross section of the declared factor. In particular, no exponential decay law, no lifetime $ℏ / Γ$ and no Lorentzian line shape of a measured distribution is asserted or derived; those would be separate models with their own assumptions and their own uncertainty qualifications.

Remarks

Relation to the full cross section. Substituting the declared channel factor into the phase-shift series of Partial-wave expansion, unit-modulus channel factors and the phase-shift cross-section series produces a conditional total cross section for an idealized situation in which a single channel dominates; the pair does not assert that any declared potential has that behavior.
Width versus rate. The width parameter $Γ$ is an energy parameter of the declared factor, not a decay rate; the identification of $Γ$ with a rate times $ℏ$ requires a separate decay model and is not made here.

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Partial-wave expansion, unit-modulus channel factors and the phase-shift cross-section series

Physical scope

The declared short-range scattering model with a real bounded central compactly supported potential in the pair's stationary elastic beam reading: rotational reduction of the amplitude to a function of the scattering angle, its convergent Legendre expansion, the unit-modulus channel factors, the phase shift read modulo pi, and the total cross section as a convergent phase-shift series in m^2. Conditional on the declared model and the cited mathematics; no measured angular distribution, no low-energy extrapolation beyond the stated series and no channel identification for material systems is asserted.

Statement

Adopt the declared short-range scattering model The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units with mass $m > 0$ , spin zero and a declared central short-range potential $V (x) = v (∣ x ∣)$ in the sense of The model's partial-wave and phase-shift vocabulary for a declared central potential clause (a): $v$ is real, bounded and $V$ is supported in a ball $B_{R}$ . Adopt the stationary elastic beam model at wave number $k > 0$ and direction $d \in S^{2}$ , with amplitude $f_{k}$ as in The model's stationary scattering wave, its outgoing amplitude and the differential and total cross-section readings, and the channel vocabulary of The model's partial-wave and phase-shift vocabulary for a declared central potential clauses (b), (c). Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Then:

(a) $f_{k}$ depends only on $\cos θ = Ω \cdot d$ , and its Legendre expansion $f_{k} (θ) = \frac{1}{2 i k} \sum_{l \geq 0} (2 l + 1) (S_{l} (k) - 1) P_{l} (\cos θ)$ converges in $L^{2} (S^{2})$ and absolutely, so the model's cross-section reading is $\frac{d σ}{d Ω} (θ) = ∣ f_{k} (θ) ∣^{2};$

(b) the channel factors are unit modulus, $∣ S_{l} (k) ∣ = 1$ , the phase shifts $δ_{l} (k)$ are real modulo $π$ , and the total cross section of the beam model is $σ = \frac{π}{k^{2}} \sum_{l \geq 0} (2 l + 1) ∣ S_{l} (k) - 1 ∣^{2} = \frac{4 π}{k^{2}} \sum_{l \geq 0} (2 l + 1) \sin^{2} δ_{l} (k) in m^{2},$ the series converging, with the channel readings of The model's partial-wave and phase-shift vocabulary for a declared central potential clause (c) as its summands;

(c) the objects $φ_{l}$ , $f_{l}$ , $S_{l}$ and $δ_{l}$ of the channel vocabulary are exactly the regular and outgoing radial solutions, the Wronskian ratio and the phase shift constructed by the cited mathematics for the reduced radial potential $w = (2 m / ℏ^{2}) v$ at the model's wave number $k$ ; the phase shifts are not asserted to have a definite sign and no low-energy limit is asserted beyond the convergence of the series.

The statement is conditional on the declared central model, the beam model and the cited mathematics; it asserts no measured angular distribution and no identification of $l$ with a measured angular momentum.

Facts & Assumptions

Given: the declared central short-range potential $V = v (∣ \cdot ∣)$ supported in $B_{R}$ , the wave number $k > 0$ , the direction $d$ , the amplitude $f_{k}$ of the stationary theorem, and the channel vocabulary of the pair.

[F1]

Partial-wave cross-section convergence. Assume Countable Choice. For real bounded central $V_{0}$ supported in $B_{R}$ and $k > 0$ , with $f$ the amplitude of the stationary outgoing solution, $f$ depends only on $\cos θ$ , its Legendre expansion with $S_{l}$ equals $\overline{W (φ_{l}, f_{l})} / W (φ_{l}, f_{l}) = e^{2 i δ_{l}}$ converges in $L^{2} (S^{2})$ and absolutely, $∣ S_{l} ∣ = 1$ , and the total cross section equals the phase-shift series (Partial-wave cross-section convergence clauses (a), (b)).

[F2]

The model's amplitude and cross-section reading. The stationary theorem of the pair constructs the model's amplitude $f_{k}$ and reads $d σ / d Ω = ∣ f_{k} ∣^{2}$ and $σ = \int ∣ f_{k} ∣^{2}$ , and the channel vocabulary declares $φ_{l}, f_{l}, S_{l}, δ_{l}$ and $σ_{l}$ (The model's stationary scattering wave, its outgoing amplitude and the differential and total cross-section readings clauses (b), (c), The model's partial-wave and phase-shift vocabulary for a declared central potential clauses (b), (c)).

[F3]

The energy-divided equation in physical coordinates. Dividing the physical stationary equation by $c = ℏ^{2} / (2 m)$ gives the energy-divided equation with the reduced central potential $W = (2 m / ℏ^{2}) V = v_{W} (∣ \cdot ∣)$ , with $W$ and $k^{2}$ in inverse-square-length units; the numerical mathematical problem uses the same $k$ (Unit, time and spectral correspondence between the physical scattering operators and the nondimensional Laplacian clauses (a), (b)).

Proof

technique · direct; reduce the physical stationary problem to its energy-divided form in physical coordinates, note that centrality is preserved, and instantiate the cited partial-wave theorem, whose amplitude is the model's amplitude

Given: the data and hypotheses of the statement and [F1]-[F3].

1.1F3givenalgebra

By [F3] the physical stationary problem at wave number $k$ and reduced potential $W = (2 m / ℏ^{2}) V$ is the energy-divided problem with $W (x) = w (∣ x ∣)$ , $w = (2 m / ℏ^{2}) v$ , which is real, bounded and supported in $B_{R}$ ; hence $W$ is a central potential of the class of [F1].

2.1F1F2step 1.1

By [F2] the model's amplitude $f_{k}$ is the amplitude of the stationary outgoing solution of the energy-divided problem of step 1.1, so [F1] gives that $f_{k}$ depends only on $\cos θ$ , that its Legendre expansion has the displayed coefficients $S_{l}$ and converges in $L^{2} (S^{2})$ and absolutely, and by [F2] the model reads $d σ / d Ω = ∣ f_{k} ∣^{2}$ . This is (a).

3.1F1F2step 2.1algebra

By [F1] the factors satisfy $S_{l} = e^{2 i δ_{l}}$ with $∣ S_{l} ∣ = 1$ for real $δ_{l}$ modulo $π$ , and the total cross section equals the displayed phase-shift series; substituting the model's channel readings $σ_{l} = (4 π / k^{2}) (2 l + 1) \sin^{2} δ_{l}$ from [F2] exhibits the series as the sum of the channel readings, and its convergence is [F1]'s. This is (b).

4.1F1step 1.1F2algebra∎

Since the energy-divided problem of step 1.1 has the same reduced radial equation and the same wave number $k$ as the model's channel vocabulary, the functions $φ_{l}, f_{l}$ and the Wronskian ratio $S_{l} = \overline{W (φ_{l}, f_{l})} / W (φ_{l}, f_{l})$ of the vocabulary are exactly those constructed by [F1]; the phase shift is the half-argument $δ_{l}$ modulo $π$ by definition of the vocabulary, and no sign or low-energy statement is added. This is (c) and completes the proof.

Remarks

Resonance models. A single-channel model of the phase shift is declared and analysed separately on this page (The declared single-channel resonance model: pole-shaped channel factor and its cross-section reading); this theorem asserts nothing about the existence of resonances for a given potential.
Low-energy behaviour. The series converges for every $k > 0$ of the declared class; no statement is made about its behaviour as $k \to 0$ , where the class hypotheses themselves remain in force but the pair asserts no limiting formula.

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The controlled Born reading: operator-norm condition, uniform amplitude error and coupling scaling

Physical scope

The declared short-range scattering model with a real bounded compactly supported potential in the pair's stationary elastic beam reading: the controlled weak-coupling condition q<1 in the operator norm of the outgoing comparison operator, the Neumann distance of the exact stationary solution from the incident wave, the uniform error of the Born amplitude, and the quadratic-in-coupling scaling form for a real or complex coupling multiplier eta. Conditional on the declared model and the cited mathematics; the item asserts no generic weak-potential approximation, no measured cross section and no validity of the Born reading outside the stated operator-norm condition.

Statement

Adopt the declared short-range scattering model The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units with mass $m > 0$ , spin zero and a compact-support-class potential $V$ supported in $B_{R}$ , adopt the stationary elastic beam model at wave number $k > 0$ and direction $d \in S^{2}$ , and let $W : = \frac{2 m}{ℏ^{2}} V, G_{k} (z) = \frac{e^{i k ∣ z ∣}}{4 π ∣ z ∣}, (K_{k} f) (x) = \int_{B_{R}} G_{k} (x - y) W (y) f (y) d y$ and $u = (I + K_{k})^{- 1} e_{i n c}$ , $f_{k}$ be as in The model's stationary scattering wave, its outgoing amplitude and the differential and total cross-section readings, with $e_{i n c} = e^{i k d \cdot x} ∣_{B_{R}}$ . Put $q : = ∥ K_{k} ∥$ , the operator norm of The operator norm as the least bound and as the unit-sphere or unit-ball supremum and A bounded linear operator between normed spaces, and $f_{B} (Ω) : = - \frac{1}{4 π} \int_{B_{R}} e^{i k (d - Ω) \cdot y} W (y) d y$ for $Ω \in S^{2}$ , read in metres. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Then:

(a) the operator $K_{k}$ is bounded, $q < \infty$ , and if $q < 1$ the exact stationary solution satisfies $∥ u - e_{i n c} ∥_{L^{2} (B_{R})} \leq \frac{q}{1 - q} ∥ e_{i n c} ∥_{L^{2} (B_{R})};$

(b) under the same condition $q < 1$ the Born amplitude $f_{B}$ approximates the exact amplitude uniformly, $∣ f_{k} (Ω) - f_{B} (Ω) ∣ \leq \frac{∥ W ∥_{\infty} ∣ B_{R} ∣^{1 / 2}}{4 π} \frac{q}{1 - q} ∥ e_{i n c} ∥_{L^{2} (B_{R})} (Ω \in S^{2}),$ with $∣ B_{R} ∣ = 4 π R^{3} / 3$ ;

(c) coupling scaling. For a real or complex multiplier $η$ with $∣ η ∣ q < 1$ , let $u_{η} : = (I + η K_{k})^{- 1} e_{i n c}$ , let $f_{k, η} (Ω; d) : = - η (4 π)^{- 1} \int_{B_{R}} e^{- i k Ω \cdot y} W (y) u_{η} (y; d) d y$ be the amplitude for the scaled potential $η W$ , and let $f_{B}$ be as above. Then $∥ u_{η} - e_{i n c} ∥_{2} \leq \frac{∣ η ∣ q}{1 - ∣ η ∣ q} ∥ e_{i n c} ∥_{2}, \sup_{Ω, d \in S^{2}} ∣ f_{k, η} (Ω; d) - η f_{B} (Ω; d) ∣ \leq \frac{∣ η ∣^{2} ∥ W ∥_{\infty} ∣ B_{R} ∣ q}{4 π (1 - ∣ η ∣ q)};$ in particular no condition on the base operator norm of $K_{k}$ itself is needed for (c).

The statement is conditional on the declared model and the cited mathematics; the condition $q < 1$ is an operator-norm condition on the declared potential, kernel and wave number, not a vague smallness of $V$ , and no measured cross section is asserted.

Facts & Assumptions

Given: the declared model and beam model with $m, ℏ > 0$ , compact-support-class $V$ supported in $B_{R}$ , wave number $k > 0$ , direction $d$ , the reduced potential $W$ , kernel $G_{k}$ , operator $K_{k}$ , exact solution $u$ and amplitude $f_{k}$ of the stationary theorem, Born amplitude $f_{B}$ , $q = ∥ K_{k} ∥$ , and the choice hypotheses.

[F1]

Controlled Born amplitude error. Assume Countable Choice. Adopt the hypotheses and notation of the nondimensional stationary outgoing integral theorem for a real bounded potential $V_{0}$ supported in $B_{R}$ : $q = ∥ K_{k} ∥$ , $u = (I + K_{k})^{- 1} e_{i n c}$ and $f_{k}$ the amplitude. If $q < 1$ , then $∥ u - e_{i n c} ∥_{2} \leq \frac{q}{1 - q} ∥ e_{i n c} ∥_{2}$ and $∣ f_{k} (Ω) - f_{B} (Ω) ∣ \leq \frac{∥ V_{0} ∥_{\infty} ∣ B_{R} ∣^{1 / 2}}{4 π} \frac{q}{1 - q} ∥ e_{i n c} ∥_{2}$ uniformly in $Ω$ ; for $V_{η} = η V_{0}$ with $∣ η ∣ q < 1$ the scaling displays hold with $u_{η} = (I + η K_{k})^{- 1} e_{i n c}$ and the same amplitude integrals, for real and complex $η$ , with no base condition on $q$ (Controlled Born amplitude error clauses (a), (b), (c)).

[F2]

The model's stationary solution. The model's $u$ , $Ψ$ and amplitude $f_{k}$ are those of the stationary theorem of the pair, constructed from the reduced potential $W = (2 m / ℏ^{2}) V$ and the same kernel and operator $K_{k}$ (The model's stationary scattering wave, its outgoing amplitude and the differential and total cross-section readings clauses (a), (b), Unit, time and spectral correspondence between the physical scattering operators and the nondimensional Laplacian clauses (a), (b)).

Proof

technique · direct; instantiate the abstract controlled Born theorem with the energy-divided physical-coordinate potential $W=W_x$ (in $\mathrm{m^{-2}}$), retaining physical $k$, $R$ and the amplitude in metres, then rewrite its constants in the model's symbols

Given: the data and hypotheses of the statement and [F1]-[F2].

1.1F1F2given

The reduced potential $W = (2 m / ℏ^{2}) V$ is real, bounded and supported in $B_{R}$ , and by [F2] the model's $K_{k}$ , $u$ and $f_{k}$ are exactly the objects of [F1] for $V_{0} = W$ ; hence [F1] applies with $q = ∥ K_{k} ∥ < \infty$ .

2.1F1step 1.1algebra

If $q < 1$ , [F1] gives $∥ u - e_{i n c} ∥_{2} \leq \frac{q}{1 - q} ∥ e_{i n c} ∥_{2}$ , which is (a).

2.2F1step 1.1algebra

Under the same condition, [F1]'s uniform estimate with $V_{0} = W$ is $∣ f_{k} (Ω) - f_{B} (Ω) ∣ \leq \frac{∥ W ∥_{\infty} ∣ B_{R} ∣^{1 / 2}}{4 π} \frac{q}{1 - q} ∥ e_{i n c} ∥_{2}$ for every $Ω \in S^{2}$ , which is the display of (b) since $∣ B_{R} ∣ = 4 π R^{3} / 3$ for the ball.

3.1F1step 2.2algebra∎

For real or complex $η$ with $∣ η ∣ q < 1$ , [F1]'s scaling clause gives the two displays of (c) with $u_{η} = (I + η K_{k})^{- 1} e_{i n c}$ and the same amplitude integrals; the constants are exactly $∥ W ∥_{\infty}$ , $∣ B_{R} ∣$ and $q$ , and no base smallness condition beyond $∣ η ∣ q < 1$ is used. This completes the proof.

Remarks

Reading the condition. In the model's variables the condition of (a)-(b) is the operator-norm condition $q = ∥ K_{k} ∥ < 1$ on the declared potential, radius and wave number; the examples of this pair verify it for a uniform ball potential through an explicit Hilbert-Schmidt bound of the kernel, so no vague smallness of the potential is used.
Complex coupling. The scaling form is algebraic in $η$ and does not use the real-potential flux conclusions; for complex $η$ the pair asserts no unitarity or cross-section reading of the resulting amplitude beyond the displayed estimate.

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Identification of the model's scattering operator with the on-shell stationary operator, and the central per-channel form

Physical scope

The declared short-range scattering model with a real bounded compactly supported potential: identification of the model's dynamical scattering operator, read as the product of the wave operators, with the on-shell stationary operator built from the model's stationary amplitude, including the free energy-angular diagonalization, the rotational covariance, and the diagonal per-channel form with the unit-modulus channel factors for a central potential. Conditional on the declared model and the cited mathematics; the identification is an operator statement, not a measurement statement, and no cross-section observation is asserted.

Statement

Adopt the declared short-range scattering model The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units with mass $m > 0$ , spin zero and a compact-support-class potential $V$ supported in $B_{R}$ ; by Asymptotic completeness and the unitary scattering operator for real bounded compactly supported potentials the model's wave operators exist and $S = Ω_{+}^{*} Ω_{-}$ is unitary on $H_{0}$ and commutes with the free flow. Write $σ$ for surface measure on $S^{2}$ and $h = L^{2} (S^{2}, d σ)$ , and for every $d \in S^{2}$ let $f_{k} (\cdot, d)$ be the model's stationary amplitude of The model's stationary scattering wave, its outgoing amplitude and the differential and total cross-section readings at incident direction $d$ , with the operator $A_{k}$ on $h$ defined by $(A_{k} a) (Ω) : = \int_{S^{2}} f_{k} (Ω, d) a (d) d σ (d) .$ Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Then:

(a) the free flow is diagonalized by the free trace transform of the cited mathematics, a unitary $F_{0} : H_{0} \to L^{2} ((0, \infty), d E; h)$ with $(F_{0} g) (E, ω) = E^{1 / 4} 2^{- 1 / 2} \hat{g} (\sqrt{E} ω)$ in the energy-divided variables of the scaling correspondence, where this auxiliary $E = k^{2}$ has units $m^{- 2}$ and the physical energy is $(ℏ^{2} / 2 m) E$ , and the model's scattering operator is carried into multiplication, $F_{0} S F_{0}^{*} = M_{S (E)};$

(b) for almost every $E = k^{2} > 0$ the on-shell operator is $S (E) = I + \frac{i k}{2 π} A_{k},$ a unitary operator on $h$ , and the model's stationary amplitude $f_{k} (Ω, d)$ is the kernel of $A_{k}$ ;

(c) for $R \in S O (3)$ let $V_{R} (x) : = V (R^{- 1} x)$ and $(U_{R} a) (ω) : = a (R^{- 1} ω)$ ; then $S [V_{R}] (E) = U_{R} S [V] (E) U_{R}^{- 1};$

(d) if $V$ is in addition central, the fiber decomposes in the spherical harmonic basis, with integer labels $l \geq 0$ and $- l \leq \overset{ˉ}{m} \leq l$ , as $S (E) Y_{l \overset{ˉ}{m}} = S_{l} (k) Y_{l \overset{ˉ}{m}}$ , where $S_{l}$ and $δ_{l}$ are the model's channel factor and phase shift of The model's partial-wave and phase-shift vocabulary for a declared central potential, equal to the Wronskian ratio $\overline{W (φ_{l}, f_{l})} / W (φ_{l}, f_{l})$ , and the cross-section reading of Partial-wave expansion, unit-modulus channel factors and the phase-shift cross-section series is the phase-shift series of that channel decomposition.

The statement identifies operators and operators-valued functions of the declared model; it asserts no measurement of $S$ or $S (E)$ , no detector model and no empirical claim.

Facts & Assumptions

Given: the declared model with $m, ℏ > 0$ , compact-support-class $V$ supported in $B_{R}$ ; the model's wave operators, scattering operator and stationary amplitude; the free trace transform $F_{0}$ ; the choice hypotheses.

[F1]

Stationary and dynamical sector identification. Assume Countable Choice. For real bounded compactly supported $V_{0}$ , with $H_{0} = - Δ$ , $H = H_{0} + M_{V_{0}}$ and $S = Ω_{+}^{*} Ω_{-}$ its complete scattering operator, the free trace transform $F_{0} g (E, ω) = E^{1 / 4} 2^{- 1 / 2} \hat{g} (\sqrt{E} ω)$ is unitary onto $L^{2} ((0, \infty), d E; h)$ , $F_{0} S F_{0}^{*} = M_{S (E)}$ with $S (E) = I + \frac{i k}{2 π} A_{k}$ unitary, $A_{k}$ built from the stationary amplitude, $S [V_{R}] (E) = U_{R} S [V] (E) U_{R}^{- 1}$ , and for radial $V_{0}$ the fiber is diagonal with $S_{l} (E) = \overline{W_{l}} / W_{l}$ , $∣ S_{l} ∣ = 1$ (Stationary and dynamical scattering in the full energy-angular representation clauses (a), (b), (c), (d)).

[F2]

The model's amplitude and $S$ . The model's amplitude $f_{k} (\cdot, d)$ is the stationary theorem's amplitude for the reduced potential $W = (2 m / ℏ^{2}) V$ , and the model's $S = Ω_{+}^{*} Ω_{-}$ is the unitary operator of the compact-support completeness theorem (The model's stationary scattering wave, its outgoing amplitude and the differential and total cross-section readings clause (b), Asymptotic completeness and the unitary scattering operator for real bounded compactly supported potentials clause (c)).

[F3]

The channel vocabulary. For a central short-range potential the model declares $S_{l} = e^{2 i δ_{l}} = \overline{W (φ_{l}, f_{l})} / W (φ_{l}, f_{l})$ and $σ_{l} = (4 π / k^{2}) (2 l + 1) \sin^{2} δ_{l}$ (The model's partial-wave and phase-shift vocabulary for a declared central potential clause (c)).

Proof

technique · direct; represent physical-coordinate $W=W_x$, $k$ and $E=k^2$ numerically in the fixed metre convention; the model's operators and amplitude are the cited energy-divided ones through the scaling correspondence, so the energy-divided identification theorem applies and supplies all four clauses

Given: the data and hypotheses of the statement and [F1]-[F3].

1.1F1F2given

The reduced potential $W = (2 m / ℏ^{2}) V$ is real, bounded and supported in $B_{R}$ , so [F1] applies to it; by the scaling correspondence the model's operators $H^{f r e e}, H$ are $c A_{0}, c A$ with $A_{0} = - Δ$ , $A = - Δ + M_{W}$ and the physical and energy-divided wave operators coincide.

2.1F1F2step 1.1

By [F2] the model's stationary amplitude for direction $d$ is the amplitude of [F1]'s stationary problem for $W$ at the same wave number $k$ ; hence $A_{k}$ is exactly the on-shell amplitude operator of [F1].

3.1F1step 1.1step 2.1algebra

By [F1] the free trace transform $F_{0}$ diagonalizes the free flow and $F_{0} S F_{0}^{*} = M_{S (E)}$ with $S (E) = I + \frac{i k}{2 π} A_{k}$ ; since the model's $S$ and $A_{k}$ are the same operators as in [F1] by steps 1.1 and 2.1, the displays of (a) and (b) follow, including unitarity of the on-shell operator.

4.1F1step 3.1algebra

By [F1] the covariance identity $S [V_{R}] (E) = U_{R} S [V] (E) U_{R}^{- 1}$ holds for the energy-divided potential; since rotations act on $W$ exactly as on $V$ through $R^{- 1}$ , the same identity is the model's (c).

5.1F1F3step 4.1algebra∎

If $V$ is central, so is $W$ ; [F1], with its integer harmonic index renamed $\overset{ˉ}{m}$ as in [F3], gives $S (E) Y_{l \overset{ˉ}{m}} = S_{l} (k) Y_{l \overset{ˉ}{m}}$ with $S_{l} = \overline{W_{l}} / W_{l}$ ; by [F3] the model's channel vocabulary names exactly these objects $S_{l} = \overline{W (φ_{l}, f_{l})} / W (φ_{l}, f_{l})$ and $δ_{l}$ , and the phase-shift series of the partial-wave theorem is the total cross-section reading of this decomposition. This is (d) and completes the proof.

Remarks

What the identification is. The identity $F_{0} S F_{0}^{*} = M_{S (E)}$ is an operator identity for the declared model; it is the precise sense in which the pair may speak of "the S-matrix at physical energy $(ℏ^{2} / 2 m) E$ ", and it makes no claim that any apparatus measures $S (E)$ .
Non-central potentials. For a non-central potential no individual spherical-harmonic channel is asserted; only the covariance of (c) holds.

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The declared long-range Coulomb model and its Dollard-modified free comparison

Physical scope

The declared long-range Coulomb scattering model of the pair: the spin-zero massive state space, the one-centre Coulomb potential with coupling kappa read in joule metres and its assigned energy observable, and the Dollard-modified free comparison with its logarithmic phase, reference time and modified wave operators. The model is an additional assumption of the pair; the reference time is a declared convention that changes only asymptotic channel phases, and the definition asserts no Rutherford amplitude, no measured cross section and no identification of kappa with measured charge data.

Definition

(b) The declared Coulomb model. As an additional model assumption of the pair, fix a coupling $κ > 0$ , read in $J m$ , and adopt the one-centre Coulomb model of The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions in its spin-zero one-particle reading: the declared potential is $V_{κ} (x) : = - \frac{κ}{∣ x ∣} (x \neq 0),$ read in joules, with the value at the origin a declared convention that does not affect the theory, and the assigned energy observable is $H_{κ} = - \frac{ℏ^{2}}{2 m} Δ_{w} - \frac{κ}{∣ \cdot ∣}$ on the domain $H^{2} (R^{3}; C) = W^{2, 2} (R^{3}; C)$ , self-adjoint there by The one-centre Coulomb model: energy observable, flow, stationary states and spectral support clause (a) with flow $e^{- i t H_{κ} / ℏ}$ in the reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution. The model declares the free comparison operator to be the free assigned energy observable $H^{f r e e}$ of The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units clause (c), and reads its Coulomb coupling in the inverse-length form $γ : = \frac{2 m κ}{ℏ^{2}} (m^{- 1}), so that H_{κ} = c (- Δ_{w} - \frac{γ}{∣ \cdot ∣}), c = \frac{ℏ^{2}}{2 m} .$ The pure Coulomb potential is not in the short-range classes of The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units clause (b); it is the model's long-range case.

(c) The declared Dollard-modified comparison. Choose a reference length $ℓ_{⋆} > 0$ , read in metres, as a comparison convention, and put $t_{⋆} : = \frac{2 m ℓ_{⋆}^{2}}{ℏ} (s) .$ and, for $t \neq 0$ , let $h_{t} : [0, \infty) \to C$ be the bounded Borel function $h_{t} (x) : = \exp (i sign (t) \frac{m κ}{ℏ \sqrt{2 m x}} \log \frac{∣ t ∣}{t_{⋆}}) (x > 0), h_{t} (0) : = 1.$ The model declares the Dollard-modified free comparison to be the family $U_{D} (t) : = e^{- i t H^{f r e e} / ℏ} h_{t} (H^{f r e e}) (t \neq 0), U_{D} (0) : = I,$ where $h_{t} (H^{f r e e})$ is the bounded functional calculus (Bounded borel pvm integral clause 1) of the self-adjoint free operator: since $h_{t}$ is unimodular, $U_{D} (t)$ is unitary for every $t$ , and both factors are functions of $H^{f r e e}$ and commute. The modified wave operators of the model are the strong limits $Ω_{D \pm} : = {s-lim}_{t \to \pm \infty} e^{i t H_{κ} / ℏ} U_{D} (t),$ declared when both exist, and the modified scattering operator is $S_{D} : = Ω_{D +}^{*} Ω_{D -}$ when both exist and are isometries. The reference length and the resulting time $t_{⋆}$ are declared conventions. More exactly, changing the reference time from $t_{⋆}$ to $t_{⋆}^{'}$ multiplies $U_{D} (t)$ on the right, for each sign $σ = sign t$ , by the fixed free-energy unitary $Q_{σ} = q_{σ} (H^{f r e e}), q_{σ} (E) = \exp (- i σ \frac{m κ}{ℏ \sqrt{2 m E}} \log \frac{t_{⋆}^{'}}{t_{⋆}}) (E > 0), q_{σ} (0) = 1.$ It is a bounded unimodular Borel multiplier and commutes with the free flow. Hence, when the wave operators exist, $Ω_{D \pm}^{'} = Ω_{D \pm} Q_{\pm}$ and $S_{D}^{'} = Q_{+}^{*} S_{D} Q_{-}$ . The completeness and isometry properties are unchanged; these are the fixed outgoing and incoming phase conventions.

(d) Status and scope. Clauses (b)-(c) are declarations of the pair: a prescribed long-range potential with its assigned energy observable, and a modified comparison construction for it. The model asserts no formula for a Coulomb scattering amplitude, no Rutherford cross section and no measured quantity; the modification is a comparison construction for the long-range generator, not a new interaction or a measured force. The ordinary short-range comparison of The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units clause (d) is not claimed to be complete for the Coulomb model; its completeness is established only for the modified comparison, on this page.

Remarks

Why a modified comparison is needed. The free comparison $e^{i t H_{κ} / ℏ} e^{- i t H^{f r e e} / ℏ}$ does not have a complete strong limit in general for a $1 / ∣ x ∣$ tail: the classical phase mismatch of the long-range interaction is not integrable in time. The logarithmic phase of $h_{t}$ is the declared correction; the model does not derive it from a measured force law.
Units of the phase. The exponent is dimensionless: $m κ / (ℏ \sqrt{2 m x})$ is dimensionless for $x$ in joules, and $\log (∣ t ∣ / t_{⋆})$ is dimensionless for $t$ in seconds and $t_{⋆} = 2 m ℓ_{⋆}^{2} / ℏ$ in seconds. Indeed $m / ℏ$ has dimension $s m^{- 2}$ , and the reference length squared supplies the missing $m^{2}$ . The quantity $γ = 2 m κ / ℏ^{2}$ has inverse-length units; its numerical coefficient becomes dimensionless only after a length unit is fixed.

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Completeness of the Dollard-modified wave operators for the declared Coulomb model

Physical scope

The declared long-range Coulomb model with coupling kappa>0 and its Dollard-modified free comparison: existence of the modified wave operators as strong limits, their isometry and intertwining, the completeness equalities of their ranges with the absolutely continuous and continuous subspaces of the Coulomb operator, and unitarity of the modified scattering operator. Conditional on the declared model and the cited mathematics; the item asserts no Coulomb scattering amplitude, no Rutherford cross section, no measured quantity and no ordinary short-range comparison for the Coulomb potential.

Statement

Adopt the declared long-range Coulomb model The declared long-range Coulomb model and its Dollard-modified free comparison with mass $m > 0$ , spin zero, action constant $ℏ > 0$ , coupling $κ > 0$ , the assigned energy operator $H_{κ}$ on $H^{2} (R^{3}; C)$ , the free operator $H^{f r e e}$ , the inverse-length coupling $γ = 2 m κ / ℏ^{2}$ , a declared reference length $ℓ_{⋆} > 0$ in metres and reference time $t_{⋆} = 2 m ℓ_{⋆}^{2} / ℏ$ , the Dollard-modified comparison $U_{D} (t)$ and the modified wave operators $Ω_{D \pm}$ and $S_{D}$ of that definition. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Then:

(a) the modified wave operators exist as strong limits and are bounded linear isometries, $Ω_{D \pm}^{*} Ω_{D \pm} = I$ , and they intertwine, $Ω_{D \pm} e^{- i s H^{f r e e} / ℏ} = e^{- i s H_{κ} / ℏ} Ω_{D \pm} (s \in R);$

(b) the modified wave operators are complete, $ran Ω_{D +} = ran Ω_{D -} = H_{a c} (H_{κ}) = H_{c} (H_{κ});$

(c) the modified scattering operator $S_{D} = Ω_{D +}^{*} Ω_{D -}$ is unitary on $H_{0}$ and commutes with the free flow;

(d) the ordinary short-range comparison of the model is not substituted: neither existence nor completeness of ${s-lim}_{t \to \pm \infty} e^{i t H_{κ} / ℏ} e^{- i t H^{f r e e} / ℏ}$ is asserted, and the modification is essential to the statement.

The statement is conditional on the declared model and the cited mathematics. It asserts no explicit Coulomb scattering amplitude, no Rutherford or other cross section, and no measured quantity; the logarithmic phase is a comparison construction for the long-range generator.

Facts & Assumptions

Given: the declared Coulomb model with $m, ℏ > 0$ , spin zero, $κ > 0$ , $γ = 2 m κ / ℏ^{2}$ , $c = ℏ^{2} / (2 m)$ , $ℓ_{⋆} > 0$ , $t_{⋆} = 2 m ℓ_{⋆}^{2} / ℏ$ , the operators $H_{κ} = c (- Δ_{w} - γ / ∣ \cdot ∣)$ and $H^{f r e e} = c (- Δ_{w})$ on $W^{2, 2}$ , the comparison $U_{D} (t)$ and the choice hypotheses.

[F1]

Dollard integrable remainder. Assume Countable Choice. For $γ > 0$ , with $H_{0} = - Δ$ , $H_{γ} = - Δ - γ / ∣ \cdot ∣$ on $H^{2}$ and the Dollard-modified evolution $U_{D}^{n d} (τ) = e^{- i τ H_{0}} e^{i sign (τ) a (P) \log ∣ τ ∣}$ , $a (p) = γ / (2 ∣ p ∣)$ , the family is unitary and the remainder $γ / ∣ \cdot ∣ - γ / (2 ∣ τ ∣ ∣ P ∣)$ is integrable along $U_{D}^{n d}$ : there are $T_{ψ}, C_{ψ}$ with $∥ [γ / ∣ \cdot ∣ - γ / (2 ∣ τ ∣ ∣ P ∣)] U_{D}^{n d} (τ) ψ ∥_{2} \leq C_{ψ} (1 + \log ∣ τ ∣) / ∣ τ ∣^{2}$ for $∣ τ ∣ \geq T_{ψ}$ , for vectors in the dense momentum-test class (Dollard integrable remainder clauses (a), (b)).

[F2]

Dollard channel completeness. With the hypotheses of [F1], the modified wave operators $Ω_{D \pm} = s - \lim_{τ \to \pm \infty} e^{i τ H_{γ}} U_{D}^{n d} (τ)$ exist on all of $L^{2}$ , satisfy $Ω_{D \pm} e^{- i s H_{0}} = e^{- i s H_{γ}} Ω_{D \pm}$ , act in each angular channel as multiplication by a bounded Borel function of the energy with $∣ b_{l, \pm} (E) ∣ = 1$ for a.e. $E > 0$ , and $ran Ω_{D \pm} = H_{a c} (H_{γ}) = H_{c} (H_{γ})$ ; $S_{D} = Ω_{D +}^{*} Ω_{D -}$ is unitary (Dollard channel completeness clauses (a), (b), (c)).

[F3]

Spectral correspondence of a positive multiple. For a self-adjoint operator $T$ , positive $c_{0}$ and every Borel $B$ , the projection valued measure of $c_{0} T$ satisfies $E_{c_{0} T} (B) = E_{T} (B / c_{0})$ , so $H_{a c}$ , $H_{s c}$ , $H_{p p}$ and $H_{c}$ of $c_{0} T$ and $T$ coincide; consequently for a bounded Borel $f$ one has $f (c_{0} T) = (f \circ c_{0}) (T)$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Bounded borel pvm integral, Unbounded Borel functional calculus: domains, products, spectral mapping clause 2, Positive scalar multiples of self-adjoint operators: domains, resolvents, spectra and eigenbases clauses (a), (b)).

[F4]

The model's readings. The declared comparison, modified wave operators and spectral vocabulary are those of clauses (b), (c) of the Coulomb definition; the Coulomb operator is self-adjoint on $H^{2}$ with the flow $e^{- i t H_{κ} / ℏ}$ (The declared long-range Coulomb model and its Dollard-modified free comparison clauses (b), (c), The one-centre Coulomb model: energy observable, flow, stationary states and spectral support clauses (a), (b)).

Proof

technique · direct; represent the physical data numerically in SI, compare with the supplied mathematical Dollard evolution through a fixed incoming or outgoing energy phase, and transfer the full limits, intertwining, completeness and scattering unitarity

Given: the data and hypotheses of the statement and [F1]-[F4].

1.1F3F4givenalgebra

[Numerical SI comparison with dimensionless arguments] Write the fixed SI units as $L_{0} = 1 m$ , $T_{0} = 1 s$ , $M_{0} = 1 k g$ and $E_{0} = 1 J$ . Let $m_{0} = m / M_{0}$ , $h_{0} = ℏ / (E_{0} T_{0})$ , $k_{0} = κ / (E_{0} L_{0})$ , $L = ℓ_{⋆} / L_{0} > 0$ and $t_{0} = t / T_{0}$ be their numerical values, and put $γ_{0} = γ L_{0} = 2 m_{0} k_{0} / h_{0}^{2}$ , $c_{0} = h_{0}^{2} / (2 m_{0})$ and $τ = h_{0} t_{0} / (2 m_{0})$ . These are pure real numbers, with $τ \to \pm \infty$ as $t \to \pm \infty$ . Represent the wavefunctions in the numerical SI coordinate $q = x / L_{0}$ , with the normalized $L^{2}$ coordinate map $J_{0} f (q) = L_{0}^{3 / 2} f (L_{0} q)$ , the Jacobian normalization understood in the chosen length unit. Substitution in the integral proves this map unitary; weak derivative substitution gives its exact $H^{2}$ domain transport. The numerical energy representatives are therefore $H_{κ} = E_{0}^{- 1} J_{0} H_{κ} J_{0}^{- 1} = c_{0} C_{γ_{0}}$ and $H_{f} = E_{0}^{- 1} J_{0} H^{f r e e} J_{0}^{- 1} = c_{0} A_{0}$ , where $A_{0} = - Δ_{q}$ and $C_{γ_{0}} = - Δ_{q} - γ_{0} / ∣ q ∣$ . Thus $J_{0} e^{- i t H_{κ} / ℏ} J_{0}^{- 1} = e^{- i τ C_{γ_{0}}}$ and similarly for the free operator. The numerical reference time is $t_{⋆, 0} = 2 m_{0} L^{2} / h_{0}$ , so $\frac{∣ t ∣}{t_{⋆}} = \frac{∣ t_{0} ∣}{t_{⋆, 0}} = \frac{∣ τ ∣}{L^{2}}, \log \frac{∣ t ∣}{t_{⋆}} = \log ∣ τ ∣ - 2 \log L .$ Every logarithm has a dimensionless argument; no time having units of length squared is placed inside the source logarithm.

2.1step 1.1F1F3F4algebra

[Exact fixed energy phases] For a positive numerical energy $e = c_{0} λ$ , the phase coefficient of the physical definition is $m_{0} k_{0} / (h_{0} \sqrt{2 m_{0} e}) = γ_{0} / (2 \sqrt{λ})$ . By [F1], $a (P) = γ_{0} / (2 ∣ P ∣)$ for the numerical source $A_{0} = ∣ P ∣^{2}$ . For $σ \in {+ 1, - 1}$ define the fixed bounded unitary $B_{σ, L} = b_{σ, L} (A_{0}), b_{σ, L} (λ) = \exp (- 2 i σ \frac{γ_{0}}{2 \sqrt{λ}} \log L) (λ > 0), b_{σ, L} (0) = 1.$ The functions are bounded unimodular Borel functions despite their oscillation near zero, so their calculus is defined on all $L^{2}$ and they commute with $e^{- i r A_{0}}$ for every $r$ . Spectral correspondence [F3] and step 1.1 give, for $∣ τ ∣ \geq 1$ and $σ = sign t$ , $J_{0} U_{D} (t) J_{0}^{- 1} = e^{- i τ A_{0}} e^{i σ a (P) \log ∣ τ ∣} B_{σ, L} = U_{D}^{n d} (τ) B_{σ, L} .$ The bounded functional calculus also transports under $J_{0}$ : conjugating the PVM gives a regular PVM representing the numerical operator, so spectral uniqueness identifies it and simple-function approximation gives the claimed calculus transport. If desired, the annular smooth momentum class in [F1] is preserved by each $B_{σ, L}$ because its multiplier is smooth on every compact annulus; hence the source remainder estimate remains integrable on that class after this fixed phase. The comparison for finite $∣ τ ∣ < 1$ need not equal the source's short-time extension and is irrelevant to the two asymptotic limits.

3.1step 1.1step 2.1F2F3algebra

[Strong limits, isometry and intertwining] Apply [F2] at the positive numerical coupling $γ_{0}$ . Let $W_{\pm}$ be its two full-space Dollard wave operators. Step 2.1 makes the physical comparison product, conjugated by $J_{0}$ , equal $e^{i τ C_{γ_{0}}} U_{D}^{n d} (τ) B_{\pm, L}$ on the corresponding exterior half-line. The fixed bounded right factor and the positive scalar time change let the strong limits pass on every vector, giving $Ω_{D \pm} = J_{0}^{- 1} W_{\pm} B_{\pm, L} J_{0} .$ Both $W_{\pm}$ are isometries by [F2] and both $B_{\pm, L}$ are unitaries, so these operators are isometries. For physical $s \in R$ , put $r = h_{0} (s / T_{0}) / (2 m_{0})$ . The numerical free/Coulomb flows are $e^{- i r A_{0}}$ and $e^{- i r C_{γ_{0}}}$ by step 1.1. Since $B_{\pm, L}$ commute with the free flow and $W_{\pm} e^{- i r A_{0}} = e^{- i r C_{γ_{0}}} W_{\pm}$ , conjugating gives the exact intertwining of (a). No new source existence or momentum-domain assumption was introduced.

4.1step 2.1step 3.1F2F3algebra

[Full completeness under the reference convention] Because $B_{\pm, L}$ are onto, step 3.1 gives $ran Ω_{D \pm} = J_{0}^{- 1} ran W_{\pm}$ . By [F2], this is $J_{0}^{- 1} H_{a c} (C_{γ_{0}}) = J_{0}^{- 1} H_{c} (C_{γ_{0}})$ . Positive energy scaling by $c_{0}$ preserves these spectral subspaces [F3], and the unitary numerical coordinate representation transports the spectral measures and their subspaces as in step 2.1. The resulting physical subspaces are exactly $H_{a c} (H_{κ}) = H_{c} (H_{κ})$ . This proves (b) for every declared reference length, preserving the source's whole continuous subspace and all angular channels.

5.1step 2.1step 3.1step 4.1F2F3F4algebra

[Scattering phases, unitarity and free-flow commutation] Put $S^{n d} = W_{+}^{*} W_{-}$ . The formula in step 3.1 gives $S_{D} = J_{0}^{- 1} B_{+, L}^{*} S^{n d} B_{-, L} J_{0} .$ All factors are unitary by [F2] and step 2.1, so $S_{D}$ is unitary. The commutation with the free flow follows directly from the two intertwinings: taking the adjoint of the plus identity at time $- r$ gives $e^{- i r A_{0}} W_{+}^{*} = W_{+}^{*} e^{- i r C_{γ_{0}}}$ , hence $S^{n d} e^{- i r A_{0}} = W_{+}^{*} e^{- i r C_{γ_{0}}} W_{-} = e^{- i r A_{0}} S^{n d}$ . The fixed $B_{\pm, L}$ commute with that same flow, so their scattering product does too, and conjugating proves (c). Equivalently the common range projection from step 4.1 gives both $S_{D}^{*} S_{D} = I$ and $S_{D} S_{D}^{*} = I$ . For any other reference time $t_{⋆}^{'}$ , let $Q_{σ} = q_{σ} (H^{f r e e})$ with $q_{σ} (E) = \exp (- i σ α (E) \log (t_{⋆}^{'} / t_{⋆}))$ , $α (E) = m κ / (ℏ \sqrt{2 m E})$ for $E > 0$ and $q_{σ} (0) = 1$ . The defining phases give exactly $Ω_{D \pm}^{'} = Ω_{D \pm} Q_{\pm}$ and $S_{D}^{'} = Q_{+}^{*} S_{D} Q_{-}$ . These incoming/outgoing factors are different sign conventions, not a claim that the scattering operator itself is unchanged.

6.1F4givenstep 3.1step 4.1step 5.1∎

[Scope and assembly] Steps 3.1–5.1 establish all the original modified-limit, isometry, intertwining, full completeness and scattering-unitarity conclusions with their corrected dimensional convention. No step supplies the ordinary unmodified Coulomb wave limits, so (d) retains exactly that distinction. The reference length adds only a comparison convention and its fixed channel phases; it adds no hypothesis on a state or on the Coulomb interaction, no amplitude formula and no measured cross section.

Remarks

Phases and observables. The modified comparison is not unique: changing the reference time $t_{⋆}$ changes asymptotic channel phases. The completeness equalities of (b) and the unitarity of (c) are invariant under that change, which is why the definition declares the reference time as a convention.
What would be needed for an amplitude. An explicit Coulomb scattering amplitude, and with it any Rutherford-type cross section, would require the angular asymptotic analysis of the Coulomb continuum solutions and a further normalization statement; neither is contained in the completeness theorem, and the pair does not assert one.

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The resonance model's phase-shift law, unitarity and channel cross section

Physical scope

The declared single-channel resonance model with every real resonance-centre parameter E_r and positive width Gamma: unitarity and a nonsingular phase law at positive physical energies, the full-real-axis auxiliary Lorentzian factor and its exact half-height width, and the positive-energy channel cross section with its resonance-centre value when E_r>0. Half-height locations have a physical reading only when positive. Conditional algebra of a declared model; no producing potential, decay law, lifetime or measured width or count is asserted.

Statement

Adopt the declared single-channel resonance model The declared single-channel resonance model: pole-shaped channel factor and its cross-section reading with channel label $l$ , mass $m > 0$ , action constant $ℏ > 0$ and parameters $E_{r} \in R$ , $Γ > 0$ , with the phase shift read modulo $π$ from $S_{l} = e^{2 i δ_{l}}$ and the channel cross-section reading $σ_{l} (E)$ of that definition. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Put $y : = Γ / 2 > 0$ . For every real $E$ define the auxiliary real-axis Lorentzian factor $L (E) : = \frac{y^{2}}{(E - E_{r})^{2} + y^{2}} .$ This is a mathematical extension of the positive-energy phase-shift factor; physical phase shifts, $k (E)$ and cross sections remain defined only for $E > 0$ . Then:

(a) for every $E > 0$ , $∣ S_{l} (E) ∣ = 1$ and $δ_{l} (E) = - Arg (E - E_{r} + i y) (m o d u l o π),$ where for $z = x + i y$ with $y > 0$ , $Arg z$ is the unique angle $θ \in (0, π)$ with $\cos θ = x / \sqrt{x^{2} + y^{2}}$ and $\sin θ = y / \sqrt{x^{2} + y^{2}}$ . Equivalently, for $E \neq E_{r}$ one may use $δ_{l} (E) = - \arctan (y / (E - E_{r}))$ modulo $π$ ; if $E = E_{r} > 0$ , $δ_{l} (E) = - π / 2$ modulo $π$ ;

(b) for every $E > 0$ , $\sin^{2} δ_{l} (E) = L (E)$ . On the entire real axis, $L (E)$ has value $1$ exactly at $E = E_{r}$ and value $1 / 2$ exactly at $E = E_{r} \pm Γ / 2$ . Thus the full-real-axis half-height width of this factor is $Γ$ for every real $E_{r}$ . A peak or half-height location is in the physical energy range only when that location is positive; the width statement itself concerns $L$ on $R$ ;

(c) for every $E > 0$ , the channel cross-section reading is $σ_{l} (E) = \frac{4 π}{k (E)^{2}} (2 l + 1) L (E), k (E) = \frac{\sqrt{2 m E}}{ℏ},$ read in $m^{2}$ . If $E_{r} > 0$ , its value at that physical energy is $σ_{l} (E_{r}) = \frac{4 π ℏ^{2}}{2 m E_{r}} (2 l + 1) .$ If $E_{r} \leq 0$ , this model defines no physical cross section at $E_{r}$ . No maximum statement at $E_{r}$ is made for $σ_{l}$ itself, whose energy dependence also contains the factor $k (E)^{- 2}$ ; no slow-variation assumption on that factor is required.

Every display is a conditional algebraic statement about the declared model; no potential producing $S_{l}$ , no decay law, no lifetime and no measured quantity is asserted.

Facts & Assumptions

Given: the declared single-channel resonance model with channel label $l$ , parameters $E_{r}$ , $Γ > 0$ , the phase-shift and cross-section readings, and the choice hypotheses.

[F1]

The declared channel factor. For $E > 0$ , $S_{l} (E) = \frac{E - E_{r} - i Γ / 2}{E - E_{r} + i Γ / 2}$ and $S_{l} = e^{2 i δ_{l}}$ modulo $π$ , with $σ_{l} (E) = \frac{4 π}{k (E)^{2}} (2 l + 1) \sin^{2} δ_{l} (E)$ and $k (E) = ℏ^{- 1} \sqrt{2 m E}$ (The declared single-channel resonance model: pole-shaped channel factor and its cross-section reading clause (b)).

[F2]

Resonance model algebra in the cited mathematics. For the single-channel model $S (E) = \frac{E - E_{r} - i Γ / 2}{E - E_{r} + i Γ / 2}$ with $Γ > 0$ , $∣ S (E) ∣ = 1$ for real $E$ and $\sin^{2} δ = (Γ / 2)^{2} / ((E - E_{r})^{2} + (Γ / 2)^{2})$ , and substituting into the phase-shift series gives the corresponding single-channel cross section (Partial-wave cross-section convergence clause (c)).

[F3]

The pair's channel reading. The channel cross-section reading of the pair is the summand of the phase-shift series of the partial-wave theorem, with $l$ , $δ_{l}$ and $k$ read as in the resonance model (The model's partial-wave and phase-shift vocabulary for a declared central potential clause (c), Partial-wave expansion, unit-modulus channel factors and the phase-shift cross-section series clause (b)).

Proof

technique · direct; compute $|S_{l}|$ and the phase of the declared factor, then substitute the declared function into the model's channel reading

Given: the data and hypotheses of the statement and [F1]-[F3].

1.1F1givenalgebra

For any real $E$ , write $x : = E - E_{r}$ and $y : = Γ / 2 > 0$ . Then $x + i y \neq 0$ and the purely algebraic ratio $R (E) : = (x - i y) / (x + i y)$ has modulus one because its numerator is the conjugate of its denominator. On the physical range $E > 0$ , [F1] identifies $R (E) = S_{l} (E)$ , proving the first part of (a). Defining $R$ as an auxiliary real-axis rational formula does not assign $k$ or a physical cross section outside the range $E > 0$ .

2.1F1step 1.1algebra

With $r : = \sqrt{x^{2} + y^{2}} > 0$ and $θ : = Arg (x + i y) \in (0, π)$ , one has $x + i y = r e^{i θ}$ and $x - i y = r e^{- i θ}$ , hence $R (E) = e^{- 2 i θ}$ . For $E > 0$ the phase $δ_{l} = - θ$ modulo $π$ therefore matches [F1]. When $x \neq 0$ , $\tan θ = y / x$ and the principal arctangent gives the same angle modulo $π$ ; when $x = 0$ , $θ = π / 2$ , so the phase is $- π / 2$ with no division by zero. This proves all of (a).

3.1F2step 1.1step 2.1algebra

For $E > 0$ , $\sin^{2} δ_{l} = \sin^{2} θ = y^{2} / (x^{2} + y^{2}) = L (E)$ , in agreement with [F2]. Independently, the real-axis formula has positive denominator for every real $E$ and satisfies $0 < L (E) \leq 1$ , with equality $L (E) = 1$ exactly when $x = 0$ . Also $L (E) = 1 / 2$ if and only if $x^{2} = y^{2}$ , so its two real half-height points are $E_{r} - y$ and $E_{r} + y$ , separated by $2 y = Γ$ . Restricting a point to the physical range requires that point to be positive; the mathematical peak and width identities impose no condition such as $E_{r} > y$ . This proves (b).

4.1F1F3step 3.1algebra∎

For $E > 0$ , substituting step 3.1 into [F1], [F3] gives $σ_{l} (E) = 4 π (2 l + 1) L (E) / k (E)^{2}$ . If $E_{r} > 0$ , this formula is defined at $E = E_{r}$ , where $L (E_{r}) = 1$ and $k (E_{r})^{2} = 2 m E_{r} / ℏ^{2}$ , yielding the displayed resonance-centre value. If $E_{r} \leq 0$ , no such physical evaluation is in the model's domain and the auxiliary $L (E_{r}) = 1$ does not create one. The extra factor $k (E)^{- 2}$ remains energy dependent, so no location-of-maximum or slow-variation assertion is used. This proves (c), retaining every real $E_{r}$ and positive $Γ$ .

Remarks

Why no peak claim. The auxiliary real-axis Lorentzian factor peaks at $E_{r}$ ; that peak is physical only if $E_{r} > 0$ . The full channel reading carries the factor $k (E)^{- 2}$ as well; the pair states the full-real-axis half-height width of the Lorentzian factor, which is exactly $Γ$ , without asserting where the maximum of $σ_{l}$ lies.
Modulo $π$ . Both $\sin^{2}$ and the cross-section reading are insensitive to the modulo- $π$ ambiguity of $δ_{l}$ .

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Scope and empirical status of the three-dimensional scattering page

Physical scope

The scope and empirical status of the three-dimensional scattering page: the distinction between the page's model deductions (existence, completeness, stationary amplitude, cross-section readings, Born and resonance models) and any observation; the apparatus and statistical qualifications needed before any measured count cross section could be compared with the model quantities; and the explicit absence of measured data on this page.

Remarks

What the page establishes, and what it does not.

Model deductions. The page proves, conditionally on the declared short-range model and the cited mathematics, the existence of the wave operators, their isometry and intertwining (Existence of the model's wave operators, isometry and intertwining, and the purely absolutely continuous free spectrum), the asymptotic completeness and unitarity of the scattering operator for the compact-support and resolvent-tail classes (Asymptotic completeness and the unitary scattering operator for real bounded compactly supported potentials, Asymptotic completeness for the resolvent-tail short-range class, with the decaying-tail example and the explicit exclusion of the Coulomb tail), the stationary outgoing solution with its amplitude and the cross-section readings (The model's stationary scattering wave, its outgoing amplitude and the differential and total cross-section readings), the controlled Born approximation with its operator-norm condition (The controlled Born reading: operator-norm condition, uniform amplitude error and coupling scaling), the phase-shift cross-section series (Partial-wave expansion, unit-modulus channel factors and the phase-shift cross-section series), the on-shell operator identification (Identification of the model's scattering operator with the on-shell stationary operator, and the central per-channel form), and the conditional resonance-model algebra (The resonance model's phase-shift law, unitarity and channel cross section). Every one of these is a logical consequence of the stated postulates, model declarations, hypotheses and mathematics; none of them is an observation.
No observations. The page reports no experiment: it contains no apparatus, no calibration, no sample, no count, no uncertainty budget and no measured cross section. Nothing on the page should be read as a measurement of $V$ , $κ$ , a phase shift or a resonance parameter.
What a comparison would require. To compare the model quantities $d σ / d Ω$ or $σ$ with data one needs a declared target preparation (target density and composition), an incident fluence with its own uncertainty, a stated scattering geometry, a detector acceptance and efficiency model, and a background model, together with a declared statistical model for the counts. The equal-channel elastic flux assumptions of the beam model (The declared stationary elastic beam model: incident plane wave, scattered wave, current readings and differential and total cross sections clause (d)) must be justified for the particular apparatus; inelastic or multichannel processes lie outside the model and would require separate channels.
Interpretation. Agreement of a measured count rate with the model reading is evidence for the combined model-plus-apparatus assumptions; it is not a deduction that the declared model, its potential class or the framework describes nature. Disagreement would not by itself falsify the whole framework, and a finite dataset can be compatible with a model without confirming it.

Scope of the model declarations. The potential is prescribed, not derived; the Coulomb coupling $κ$ and the resonance parameters $E_{r}, Γ$ are model parameters, not measured constants; and the modified Dollard comparison of the Coulomb model is a declared construction whose phase convention changes only asymptotic channel phases (The declared long-range Coulomb model and its Dollard-modified free comparison clauses (c), (d)).

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Long-range, threshold and model-scope limits of the scattering page

Physical scope

The explicit limits of the scattering page: the exclusion of the Coulomb tail from the short-range classes and its separate modified-comparison treatment, the threshold obstruction for uniform compact-defect statements at zero energy and the fact that completeness itself is unaffected, the absence of any zero-energy stationary statement, and the conditional status of the resonance model. This records scope; it asserts no new mathematical consequence.

Remarks

Long-range tails. The short-range classes of The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units clause (b) exclude the pure Coulomb tail: the decaying-tail criterion of Asymptotic completeness for the resolvent-tail short-range class, with the decaying-tail example and the explicit exclusion of the Coulomb tail clause (c) requires a positive power $δ > 0$ , and the Coulomb quotient $(1 + ∣ x ∣ / L_{*})^{1 + δ} ∣ V (x) ∣$ is unbounded for every $δ > 0$ . The general resolvent-tail exclusion follows from clause (d) of that same theorem: its translated compact-bump input gives $h_{*} (r) \geq C / (2 r + R)$ with $C > 0$ and dimensionless radii $r, R$ (physical cutoff radius $L_{*} r$ ), which is not integrable. The Coulomb case is treated by the modified Dollard comparison (The declared long-range Coulomb model and its Dollard-modified free comparison), for which the ordinary short-range comparison is not claimed.
Threshold. The completeness statements of this page are made for the declared classes and do not require a zero-energy nonresonance hypothesis (Resolvent-tail short-range completeness). A different uniformity statement fails at zero energy: for compactly supported potentials the scattering defects are not compact uniformly when the energy cutoff does not vanish at $0$ (Compact scattering defects and the zero-energy obstruction). The general stationary beam and amplitude theorems use $k > 0$ and give no uniform zero-energy amplitude statement. The companion's declared square-well computation separately proves its explicit zero-energy resonance profile.
Positive eigenvalues. The absence of positive eigenvalues is part of Existence of the model's wave operators, isometry and intertwining, and the purely absolutely continuous free spectrum clause (d) for the compact-support class; no statement is made about negative or zero eigenvalues of the interacting operator, and no general bound-state inventory belongs to this page. The companion's explicit well has a separately proved local threshold branch, without a bound-state count for arbitrary potentials.
Resonances. The resonance model (The declared single-channel resonance model: pole-shaped channel factor and its cross-section reading) is a declared conditional model; the page proves no existence of a resonance for any declared potential and makes no analytic-continuation claim for the resolvent or the scattering operator.

5 · Examples, experiments and counterexamples

None yet.