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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.

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

Statement

Facts & Assumptions

Proof

Remarks

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