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Controlled Born amplitude error

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Adopt the hypotheses and notation of Stationary outgoing integral and flux: real $V$ bounded with $supp V \subseteq B_{R}$ , $k > 0$ , $d \in S^{2}$ , the compact operator $K_{k}$ , and write $q : = ∥ K_{k} ∥$ . Then:

(a) Neumann control. If $q < 1$ then $u = (I + K_{k})^{- 1} e_{i n c}$ satisfies $∥ u - e_{i n c} ∥_{L^{2} (B_{R})} \leq \frac{q}{1 - q} ∥ e_{i n c} ∥_{L^{2} (B_{R})} .$

(b) First Born amplitude. Under the same condition $q < 1$ , with $f_{B} (Ω) : = - \frac{1}{4 π} \int_{B_{R}} e^{i k (d - Ω) \cdot y} V (y) d y$ one has, uniformly in $Ω \in S^{2}$ , $∣ f_{k} (Ω) - f_{B} (Ω) ∣ \leq \frac{∥ V ∥_{\infty} ∣ B_{R} ∣^{1 / 2}}{4 π} \frac{q}{1 - q} ∥ e_{i n c} ∥_{L^{2} (B_{R})} .$

(c) Scaling form. For $V_{η} = η V$ with $∣ η ∣ q < 1$ , put $K_{k, η} = η K_{k}$ , $u_{η} = (I + η K_{k})^{- 1} e_{i n c}$ , and define the corresponding amplitudes by the same integrals. Then $∥ u_{η} - e_{i n c} ∥_{2} \leq \frac{∣ η ∣ q}{1 - ∣ η ∣ q} ∥ e_{i n c} ∥_{2},$ $\sup_{Ω, d \in S^{2}} ∣ f_{k, η} (Ω; d) - η f_{B} (Ω; d) ∣ \leq \frac{∣ η ∣^{2} ∥ V ∥_{\infty} ∣ B_{R} ∣ q}{4 π (1 - ∣ η ∣ q)} .$ This applies to every real $η$ in the original real stationary model. For complex $η$ the same Neumann construction and amplitude integrals give the same algebraic estimate, without asserting the real-potential flux conclusions. No base condition $q < 1$ is needed in (c). These give controlled weak-coupling Born estimates in the actual operator norm, uniformly in both incident and outgoing directions.

Facts & Assumptions

Given: the generic bounded compact real $V$ , $R, k > 0$ , $d, Ω \in S^{2}$ , and $e_{d} (y) = e^{i k d \cdot y}$ with $∥ e_{d} ∥_{2} = ∣ B_{R} ∣^{1 / 2}$ .

[F1]

The precise kernel and amplitude normalization are Stationary outgoing integral and flux; the actual all- $k$ inverse is Outgoing Fredholm invertibility. The local CC finite-product kernel lemma is L^2 kernel operators are bounded and compact. The concrete norm calculation uses Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma and Cauchy-Schwarz inequality for $L^{2}$ . Completeness and operator submultiplicativity are Hilbert space, A bounded linear operator between normed spaces and The operator norm as the least bound and as the unit-sphere or unit-ball supremum. Only their CC contracts are used.

Proof

1.1givenF1constructalgebra

The kernel $h_{k} (x, y) = 1_{B_{R}} (x) 1_{B_{R}} (y) e^{i k ∣ x - y ∣} V (y) / (4 π ∣ x - y ∣)$ lies in $L^{2} (R^{6})$ : polar integration of $∣ x - y ∣^{- 2}$ over $∣ x - y ∣ < 2 R$ gives $∥ h_{k} ∥_{2}^{2} \leq ∥ V ∥_{\infty}^{2} ∣ B_{R} ∣ R / (2 π)$ . The actual local kernel lemma supplies finite-product approximants with operator-norm error bounded by their kernel- $L^{2}$ error; zero extension and restriction give a bounded compact $K_{k}$ on $L^{2} (B_{R})$ and finite $q = ∥ K_{k} ∥$ . This is a concrete Euclidean CC construction, with no arbitrary Hilbert–Schmidt basis.

2.1step 1.1F1constructalgebra

Suppose $q < 1$ . Define $S_{N} f = \sum_{n = 0}^{N} (- K_{k})^{n} f$ . For $M > N$ , $∥ (S_{M} - S_{N}) f ∥_{2} \leq q^{N + 1} ∥ f ∥_{2} / (1 - q)$ . Completeness defines the unique linear limit $S f$ , with $∥ S ∥ \leq (1 - q)^{- 1}$ and $∥ S - S_{N} ∥ \leq q^{N + 1} / (1 - q)$ . The finite geometric identity $(I + K_{k}) S_{N} = S_{N} (I + K_{k}) = I - (- K_{k})^{N + 1}$ passes to the limit, so $S$ is the two-sided inverse. Consequently $u - e_{d} = \sum_{n \geq 1} (- K_{k})^{n} e_{d}$ and $∥ u - e_{d} ∥_{2} \leq q ∥ e_{d} ∥_{2} / (1 - q)$ , proving (a).

3.1step 2.1F1givenalgebra

The amplitude difference is $L_{Ω} (u - e_{d})$ for $L_{Ω} (v) = - (4 π)^{- 1} \int_{B_{R}} e^{- i k Ω \cdot y} V (y) v (y) d y$ . Cauchy–Schwarz gives $∣ L_{Ω} (v) ∣ \leq ∥ V ∥_{\infty} ∣ B_{R} ∣^{1 / 2} ∥ v ∥_{2} / (4 π)$ uniformly in $Ω$ . The absolute integrability follows from the same estimate. Inserting step 2.1 proves (b). Since $K_{k}$ is independent of $d$ and $∥ e_{d} ∥_{2} = ∣ B_{R} ∣^{1 / 2}$ , this is also uniform in $d$ . No radial specialization occurs.

4.1step 2.1step 3.1algebra∎

Linearity of the kernel gives $K_{k, η} = η K_{k}$ and $q_{η} = ∣ η ∣ q$ . Repeat step 2.1 with this operator whenever $∣ η ∣ q < 1$ , regardless of the base $q$ . The amplitude functional is $L_{η, Ω} = η L_{Ω}$ and $f_{B, η} = η f_{B}$ . Thus $f_{k, η} - η f_{B} = η L_{Ω} (u_{η} - e_{d})$ ; step 3.1 gives precisely both factors of $∣ η ∣$ in (c). The cases $q = 0$ or $η = 0$ follow from the same identities with zero error, without division by either. This proves the full claimed scaling and weak-coupling control.

Controlled Born amplitude error

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

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