The three-dimensional volume integral scattering model

Assume Countable Choice. Let $D\subset {\mathbb{R}}^3$ be a bounded Lebesgue-measurable set, let $M:=\mathrm{\mid }D\mathrm{\mid }$, let $q\in L^{\mathrm{\infty }}(D;\mathbb{C})$, and let $k\in \mathbb{R}$. Here $L^2$ and $L^{\mathrm{\infty }}$ are equivalence classes modulo null functions (The space $L^p(\mu )$ as the quotient by null functions), and $L^{\mathrm{\infty }}$ means essentially bounded with respect to Lebesgue measure (The essential supremum of a measurable function with respect to a measure). We regard $q$ as zero off $D$. The convention is the volume equation $u=f+K_{k}u,K_{k}u(x):={\int }_D{G}_k(x-y)q(y)u(y)dy,G_k(z):=\frac{e^{ik\mathrm{\mid }z\mathrm{\mid }}}{4\pi \mathrm{\mid }z\mathrm{\mid }}(z\mathrm{\ne }0)\mathrm{.}$ Set $G_k(0):=0$; when $M>0$ the diagonal in $D\times D$ is product-null, so this value is only a representative convention. The input and unknown are classes in complex $L^2(D)$, and the displayed equation is interpreted as an equality of those classes. Its formula is well defined by The volume integral operator is well defined and bounded on L2 ↗.

The associated far-field map is ${\mathcal{F}}_k[v](\theta ):=\frac{1}{4\pi }{\int }_D{e}^{-ik\theta \cdot y}q(y)v(y)dy,\theta \in S^2\mathrm{.}$ Here $S^2$ has the surface measure of The polar surface set function on the unit sphere. The map is well defined as a continuous function of $\theta$ by The volume far-field map is bounded and is the uniform far-field limit ↗.

For $x\mathrm{\notin }\bar{D}$, the exterior volume potential is $V_{k}v(x):={\int }_D{G}_k(x-y)q(y)v(y)dy\mathrm{.}$ For each $v\in L^2(D;\mathbb{C})$, the essential-supremum result gives $\mathrm{\mid }q\mathrm{\mid }\le \mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}$ almost everywhere, and Cauchy--Schwarz gives $\mathrm{\parallel }qv{\mathrm{\parallel }}_{L^1(D)}\le \mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}\sqrt{M}\mathrm{\parallel }v{\mathrm{\parallel }}_2$. Thus, for $x\mathrm{\notin }\bar{D}$, the exterior volume-potential integral is absolutely convergent: $\bar{D}$ is closed, so its complement contains a ball about $x$ disjoint from $D$.

If $M=0$, define $L^2(D)=\{0\}$ and $K_k=0$; all solution and far-field statements on that space are the zero-space statements. The substantive operator results below assume $0<M<\mathrm{\infty }$.

This is a stipulated integral-equation model with the sign convention $(-\mathrm{\Delta }-k^2)(u-f)=qu$ when the corresponding distributional Helmholtz equation is used. In the acoustic notation of Scherzer–Mindrinos, their equation $u=u_i-k^2\mathrm{\Phi }\ast ((1-n)u)$ corresponds to $q=k^2(n-1)$. No existence or uniqueness for all contrasts is asserted by this definition.