Fixed-frequency far-field data on the small-norm volume branch

Assume Countable Choice. Let $D\subset {\mathbb{R}}^3$ be a bounded Lebesgue-measurable set with $0<M:=\mathrm{\mid }D\mathrm{\mid }<\mathrm{\infty }$, and let $k>0$. Write $S^2:=\{v\in {\mathbb{R}}^3:\mathrm{\mid }v\mathrm{\mid }=1\}$ and $C_D:=\sqrt{M\mathrm{diam}(D)\mathrm{/}(4\pi )}$. For $q\in L^{\mathrm{\infty }}(D;\mathbb{C})$, extended by zero off $D$, set $G_k(z):=e^{ik\mathrm{\mid }z\mathrm{\mid }}\mathrm{/}(4\pi \mathrm{\mid }z\mathrm{\mid })$ for $z\mathrm{\ne }0$ and $G_k(0):=0$, and let $K_{q,k}v(x):={\int }_D{G}_k(x-y)q(y)v(y)dy,{\rho }_k(q):=\mathrm{\parallel }{K}_{q,k}{\mathrm{\parallel }}_{L^2(D)\to L^2(D)}\mathrm{.}$ The small-norm contrast class is ${\mathcal{U}}_{D,k}:=\{q\in L^{\mathrm{\infty }}(D;\mathbb{C}):{\rho }_k(q)<1\}\mathrm{.}$ For $d\in S^2$, put $f_{k,d}(x):=e^{ikd\cdot x}$ on $D$. For $q\in {\mathcal{U}}_{D,k}$, let $u_{q,k,d}\in L^2(D;\mathbb{C})$ denote the unique solution of $u=f_{k,d}+K_{q,k}u$. For $q\in {\mathcal{U}}_{D,k}$, define the exact far-field data family by ${\mathcal{S}}_k(q)(\theta ,d):=\frac{1}{4\pi }{\int }_D{e}^{-ik\theta \cdot y}q(y)u_{q,k,d}(y)dy,(\theta ,d)\in S^2\times S^2\mathrm{.}$

For $h\in L^{\mathrm{\infty }}(D;\mathbb{C})$, define its first Born data by ${\mathcal{B}}_{k}h(\theta ,d):=\frac{1}{4\pi }{\int }_D{e}^{-ik\theta \cdot y}h(y)e^{ikd\cdot y}dy=\frac{1}{4\pi }\widehat{h_0}\text{\hspace{0.17em}⁣}(\frac{k}{2\pi }(\theta -d)),$ where $h_0$ is the zero extension of $h$ to ${\mathbb{R}}^3$ and $\widehat{g}(\xi ):={\int }_{{\mathbb{R}}^3}g(y)e^{-2\pi iy\cdot \xi }dy$ for $g\in L^1({\mathbb{R}}^3;\mathbb{C})$. No uniqueness or stability outside ${\mathcal{U}}_{D,k}$ is asserted.

The diameter is $\mathrm{diam}(D)=\sup \{\mathrm{\mid }x-y\mathrm{\mid }:x,y\in D\}$.

The norm and dot product on ${\mathbb{R}}^3$ are the Euclidean ones.

The spaces $L^2$ and $L^{\mathrm{\infty }}$ consist of measurable functions modulo equality almost everywhere.

The $L^{\mathrm{\infty }}$ norm is the essential supremum with respect to Lebesgue measure.

The operator norm is the least bound on a bounded linear map between normed spaces.

The outgoing volume model fixes $G_k(z)=e^{ik\mathrm{\mid }z\mathrm{\mid }}\mathrm{/}(4\pi \mathrm{\mid }z\mathrm{\mid })$ and the far-field convention $e^{-ik\theta \cdot y}q(y)v(y)\mathrm{/}(4\pi )$.

The displayed volume kernel defines a bounded operator on complex $L^2(D)$.

The far-field map is continuous and is the uniform outgoing limit of the exterior volume potential.

The library Fourier transform uses $\widehat{g}(\xi )=\int g(y)e^{-2\pi iy\cdot \xi }dy$, giving the frequency $k(\theta -d)\mathrm{/}(2\pi )$.

For the strict operator norm bound, the volume equation has a unique L2 solution, which supplies the field used here.

Countable Choice is the declared axiom branch for the L2 volume model and its Neumann result.