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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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources