Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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 $k = 0$ .

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources