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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 $k > 0$ .

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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Sources