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.

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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources