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The volume far-field map is bounded and is the uniform far-field limit

Statement

Assume Countable Choice. Let $D, q, k$ be as in The three-dimensional volume integral scattering model, with $0 < ∣ D ∣ < \infty$ , and put $R_{0} > \sup_{y \in D} ∣ y ∣$ . Then $F_{k} : L^{2} (D; C) \to C (S^{2}; C)$ is linear and $∥ F_{k} v ∥_{\infty} \leq \frac{∥ q ∥_{\infty} \sqrt{∣ D ∣}}{4 π} ∥ v ∥_{2} .$ It is Lipschitz in direction with constant at most $∣ k ∣ R_{0} ∥ q ∥_{\infty} \sqrt{∣ D ∣} ∥ v ∥_{2} / (4 π)$ . For every $r \geq 2 R_{0}$ , the exterior volume potential of The three-dimensional volume integral scattering model satisfies $\sup_{θ \in S^{2}} ∣ r e^{- i k r} V_{k} v (r θ) - F_{k} v (θ) ∣ \leq \frac{2 R_{0} + ∣ k ∣ R_{0}^{2}}{4 π r} ∥ q ∥_{\infty} \sqrt{∣ D ∣} ∥ v ∥_{2} .$ Thus $F_{k} v$ is the uniform far-field amplitude under the outgoing $e^{i k r} / r$ convention.

Facts & Assumptions

Given: Countable Choice, $D, q, k$ as in the Statement, $v \in L^{2} (D; C)$ , and $R_{0} > \sup_{y \in D} ∣ y ∣$ .

[F1]

Cauchy–Schwarz gives $∥ q v ∥_{L^{1} (D)} \leq ∥ q ∥_{\infty} \sqrt{∣ D ∣} ∥ v ∥_{2}$ (The complex $L^{2}$ pairing is well-defined and satisfies Cauchy–Schwarz).

[F2]

For real $s, t$ , let $γ (r) = \exp (i (t + r (s - t)))$ on $[0, 1]$ . If $s = t$ , it is constant. Otherwise, with $z = i (t + r (s - t))$ , the complex derivative theorem The complex exponential is entire and its complex derivative is itself gives $\frac{γ (r + h) - γ (r)}{h} = i (s - t) \frac{\exp (z + i h (s - t)) - \exp z}{i h (s - t)} ⟶ i (s - t) \exp z .$ Thus $∣ γ^{'} (r) ∣ = ∣ s - t ∣$ by $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , and identifying $C$ with $R^{2}$ , If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz gives $∣ e^{i s} - e^{i t} ∣ \leq ∣ s - t ∣$ .

[F3]

The integral convention for the exterior potential and the far-field map is fixed in The three-dimensional volume integral scattering model.

Proof

Given: The data in the Statement.

1.1F1F3algebra

By [F1], the defining integral of $F_{k} v$ is absolutely convergent and $∣ F_{k} v (θ) ∣ \leq (4 π)^{- 1} ∥ q v ∥_{1}$ . Taking the supremum proves the norm bound; linearity follows from linearity of the integral.

1.2F1F2F3

For $θ, θ^{'} \in S^{2}$ , [F2] gives $∣ e^{- i k θ \cdot y} - e^{- i k θ^{'} \cdot y} ∣ \leq ∣ k ∣ ∣ y ∣ ∣ θ - θ^{'} ∣$ . Integrating against $∣ q v ∣ / (4 π)$ and using [F1] proves the stated Lipschitz bound and hence continuity in direction.

1.3givenF2algebra

Fix $r \geq 2 R_{0}$ , $θ \in S^{2}$ , and $y \in D$ , and set $s : = ∣ r θ - y ∣$ . Then $s \geq r - R_{0} \geq r / 2$ , $∣ r - s ∣ \leq R_{0}$ , and $∣ s - (r - θ \cdot y) ∣ = \frac{∣ y ∣^{2} - (θ \cdot y)^{2}}{s + r - θ \cdot y} \leq \frac{R_{0}^{2}}{r},$ because the denominator is at least $(r - R_{0}) + (r - R_{0}) \geq r$ . Therefore $∣ r / s - 1 ∣ \leq 2 R_{0} / r$ and, by [F2], $∣ e^{i k (s - r)} - e^{- i k θ \cdot y} ∣ \leq ∣ k ∣ R_{0}^{2} / r$ .

2.1F1F3step 1.3algebra∎

The integrand identity $r e^{- i k r} G_{k} (r θ - y) = (4 π)^{- 1} (r / s) e^{i k (s - r)}$ and step 1.3 give a uniform pointwise error at most $(2 R_{0} + ∣ k ∣ R_{0}^{2}) / (4 π r)$ . Integrating against $∣ q v ∣$ , applying [F1], and taking the supremum over $θ$ proves the far-field estimate.

Depends on

Used by

Cited to discharge well-definedness by The three-dimensional volume integral scattering model.

Dependency tree · two levels

74 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Otmar Scherzer and Leonidas Mindrinos, Inverse Problems: Lecture Notes, Sommersemester 2015, §5.2.2 (standard reference, not scraped)
Maximilian Kreuzer, Quantum Theory: Lecture Notes, Chapter 8, §8.1.2 (standard reference, not scraped)