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The first Born data are the Fréchet linearization at zero contrast

Statement

Assume Countable Choice and the hypotheses and notation of the fixed-frequency far-field definition. For $q \in U_{D, k}$ put $ρ = ρ_{k} (q)$ . Uniformly over all incident and observation directions, $\sup_{θ, d \in S^{2}} ∣ S_{k} (q) (θ, d) - B_{k} q (θ, d) ∣ \leq \frac{M}{4 π} ∥ q ∥_{\infty} \frac{ρ}{1 - ρ} .$ Whenever $C_{D} ∥ q ∥_{\infty} < 1$ , this implies $\sup_{θ, d \in S^{2}} ∣ S_{k} (q) (θ, d) - B_{k} q (θ, d) ∣ \leq \frac{M C_{D}}{4 π} \frac{∥ q ∥_{\infty}^{2}}{1 - C_{D} ∥ q ∥_{\infty}} .$ For each fixed $(θ, d)$ , the scalar map $q \mapsto S_{k} (q) (θ, d)$ on the open domain $U_{D, k}$ , with $L^{\infty} (D; C)$ and $C$ viewed as real Banach spaces, is Fréchet differentiable at $q = 0$ and has derivative $h \mapsto B_{k} h (θ, d)$ . The remainder bound is uniform in $(θ, d)$ ; it asserts no nonlinear uniqueness or stability result.

Statement references

The exact scattering family, small-norm domain, and first Born data formula.

The small-norm solution, data integrals, and open contrast domain are well-defined.

The operator-norm estimate $ρ_{k} (q) \leq C_{D} ∥ q ∥_{\infty}$ .

For $ρ < 1$ , the Lippmann–Schwinger solution satisfies the first partial-sum tail bound.

Fréchet differentiability means a bounded real-linear first-order approximation with remainder $o (∥ h ∥)$ .

Countable Choice makes complex $L^{\infty}$ complete.

The declared axiom for L-infinity completeness and volume-scattering suppliers.

Facts & Assumptions

Given: Countable Choice; the data in the fixed-frequency definition; a contrast $q \in U_{D, k}$ ; and directions $θ, d \in S^{2}$ .

[F1]

The definition fixes $K_{q, k}$ , $ρ_{k} (q)$ , $S_{k} (q)$ , and $B_{k} q$ on the stated quotient spaces and gives $C_{D} = \sqrt{M diam (D) / (4 π)}$ (Fixed-frequency far-field data on the small-norm volume branch).

[F2]

The volume operator is linear in $q$ and obeys $∥ K_{q, k} ∥_{L^{2} \to L^{2}} \leq C_{D} ∥ q ∥_{\infty}$ (The volume integral operator is well defined and bounded on L2).

[F3]

For $ρ < 1$ , the unique solution $u = f + K_{q, k} u$ satisfies $∥ u - \sum_{j = 0}^{N} K_{q, k}^{j} f ∥_{2} \leq ρ^{N + 1} ∥ f ∥_{2} / (1 - ρ)$ (Born truncation error for a small-norm volume integral operator).

[F4]

The complex L2 pairing is representative-independent and satisfies Cauchy–Schwarz (The complex $L^{2}$ pairing is well-defined and satisfies Cauchy–Schwarz).

[F5]

The Fourier transform convention is $\hat{g} (ξ) = \int g (y) e^{- 2 π i y \cdot ξ} d y$ for $g \in L^{1} (R^{3}; C)$ (Fourier transform on complex L1 classes).

[F6]

Under Countable Choice, $L^{\infty} (D; C)$ is complete; realification leaves its norm and metric unchanged. The same completeness fact applies to $C$ as a finite-dimensional complex L-infinity space (Complex Lp completeness and almost-everywhere subsequences, Realification preserves a complex norm and its metric).

[F7]

Between real Banach spaces, a Fréchet derivative is a bounded real-linear map whose remainder divided by the input norm tends to zero (Fréchet derivative between Banach spaces, Banach space).

[F8]

For real $t$ , $∣ e^{i t} ∣ = 1$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

[F9]

A measurable representative of the complex contrast satisfies $∣ q ∣ \leq ∥ q ∥_{\infty}$ almost everywhere (The essential supremum is attained as the least essential bound).

[F10]

The domain $U_{D, k}$ is open and contains zero (Well-definedness of the small-norm fixed-frequency scattering map).

[A1]

Countable Choice is assumed through [F3] and [F6] (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

Given: The data above.

1.1F1F5F8F9given

For any $h \in L^{\infty} (D; C)$ , [F8] gives $∣ e^{- i k θ \cdot y} e^{i k d \cdot y} ∣ = 1$ and [F9] gives $\int_{D} ∣ h ∣ \leq M ∥ h ∥_{\infty}$ . The integral definition therefore yields $∣ B_{k} h (θ, d) ∣ \leq M ∥ h ∥_{\infty} / (4 π)$ . By [F5] and the Fourier formula in [F1], this integral is the stated Born functional. It is complex-linear in $h$ , hence bounded and real-linear for every fixed pair of directions.

1.2F1F3F8given

Let $f = f_{k, d}$ . By the definition and [F8], $∣ f ∣ = 1$ on $D$ , so $∥ f ∥_{2} = \sqrt{M}$ . The case $N = 0$ of [F3] gives $∥ u_{q, k, d} - f ∥_{2} \leq ρ \sqrt{M} / (1 - ρ)$ .

2.1F1F3F4F9step 1.2

Subtracting the integral formulas in [F1] yields $S_{k} (q) (θ, d) - B_{k} q (θ, d) = \frac{1}{4 π} \int_{D} e^{- i k θ \cdot y} q (y) (u_{q, k, d} (y) - f (y)) d y .$ By [F9], $q (u_{q, k, d} - f)$ has $L^{2}$ norm at most $∥ q ∥_{\infty} ∥ u_{q, k, d} - f ∥_{2}$ . The same unit-modulus calculation as step 1.2, with $d$ replaced by $θ$ , gives $∥ e^{i k θ \cdot (\cdot)} ∥_{2} = \sqrt{M}$ . By [F4], the integral's absolute value is at most $∥ q ∥_{\infty} \sqrt{M} ∥ u_{q, k, d} - f ∥_{2} / (4 π)$ ; then [F3] and step 1.2 give the first bound, uniformly in the directions.

3.1F2step 2.1algebra

By [F2], $ρ \leq C_{D} ∥ q ∥_{\infty}$ . When $C_{D} ∥ q ∥_{\infty} < 1$ , monotonicity of $t \mapsto t / (1 - t)$ on $[0, 1)$ gives $ρ / (1 - ρ) \leq C_{D} ∥ q ∥_{\infty} / (1 - C_{D} ∥ q ∥_{\infty})$ . Substitution into step 2.1 proves the second bound.

4.1A1F1F6F7F10step 1.1step 3.1algebra∎

For nonzero $q$ with $C_{D} ∥ q ∥_{\infty} < 1$ , divide the second bound by $∥ q ∥_{\infty}$ ; the quotient is at most $\frac{M C_{D}}{4 π} \frac{∥ q ∥_{\infty}}{1 - C_{D} ∥ q ∥_{\infty}}$ , which tends to zero as $∥ q ∥_{\infty} \to 0$ . Since [F10] makes zero an interior point, the bounded real-linear map from step 1.1 satisfies the Fréchet remainder criterion [F7], with the Banach-space hypotheses in [F6]. Also $S_{k} (0) = 0$ by its definition. Thus $D S_{k} (0) (θ, d) h = B_{k} h (θ, d)$ for each fixed pair, with the same remainder estimate uniformly over the full direction family.

Remarks

At zero contrast the first Born data are $B_{k} h (θ, d) = \frac{1}{4 π} \hat{h_{0}} ⁣ (\frac{k}{2 π} (θ - d)) .$ Thus each fixed-frequency direction pair samples one Fourier frequency on the corresponding Ewald surface. The result is local at $q = 0$ and does not extend a derivative or inverse-uniqueness assertion to arbitrary contrasts.

Fourier-transfer geometry

For fixed $k > 0$ and unit directions $d, θ$ , the wavevectors $k d$ and $k θ$ lie on the radius- $k$ sphere in their span. The arrow from $k d$ to $k θ$ is exactly $k (θ - d)$ ; the Fourier variable in the library's $2 π$ convention is that chord divided by $2 π$ . The diagram is only a geometric description of the frequency label; the Fréchet estimate above is proved by the integral and norm bounds, not by the drawing.

Depends on

Used by

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58 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

Fioralba Cakoni, David Colton, and Houssem Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS 88 (standard reference, not scraped)
Otmar Scherzer and Leonidas Mindrinos, Inverse Problems: Lecture Notes, University of Vienna, Sommersemester 2015 (standard reference, not scraped)
Semyon Dyatlov, MIT 18.155 Lecture Notes: Distributions, Elliptic Regularity, and Applications to PDEs (standard reference, not scraped)