Estimating the size of the scatterer

TL;DR

This paper derives an explicit formula to estimate the size of a scatterer, such as an obstacle or potential support, based on its scattering amplitude.

Contribution

It provides a novel explicit formula linking the scatterer's size directly to measurable scattering data.

Findings

Explicit size estimation formula derived

Applicable to obstacles and potential supports

Enhances inverse scattering analysis

Abstract

Formula for the size of the scatterer is derived explicitly in terms of the scattering amplitude corresponding to this scatterer. By the scatterer either a bounded obstacle $D$ or the support of the compactly supported potential is meant

Equations

$${\nabla }^{2}u+k^{2}u=0in{D}^{\prime },u{|}_S=0,u=u_0+v,$$

$$v_r-ikv=O(r^{-2}),r=|x|\to \infty ,x/r=\beta ,$$

$$u=u_0(x,\alpha )-{\int }_{S}g(x,s)h(s)ds,h:=u_N(s,\alpha ).$$

$$u=u_0+A(\beta ,\alpha )\frac{e^{ikr}}{r}+O(r^{-2}),r\to \infty ,\beta =x/r,$$

$$A(\beta ,\alpha )=-\frac{1}{4\pi }{\int }_S{e}^{-ik\beta \cdot s}h(s,\alpha )ds.$$

$$\beta =a{e}_1+ib{e}_2,a,b\in \mathbb{R},a^2-b^2=1.$$

$$d\le \underset{b\to \infty }{\lim }\frac{\ln |A(a{e}_1+ib{e}_2,\alpha )|}{bk},$$

$$[{\nabla }^2+k^2-q(x)]\psi =0in{\mathbb{R}}^3,$$

$$A(\beta ,\alpha )=-\frac{1}{4\pi }{\int }_D{e}^{-ik\beta \cdot y}H(y)dy,H(y)=H(y,\alpha ):=q(y)\psi (y,\alpha ).$$

$$\underset{y\in D,\alpha \in S^2}{\max }|H(y,\alpha )|\le c,$$

$$|A(a{e}_1+ib{e}_2,\alpha )|\le c{\int }_S{e}^{bk{s}_2}ds,c=\frac{1}{4\pi }\underset{s\in S}{\max }|u_N(s,\alpha )|.$$

$$\underset{b\to \infty }{\lim }\frac{\ln {\int }_S{e}^{bk{s}_2}ds}{bk}=d,$$

$${\int }_S{e}^{bk{s}_2}ds={\int }_{S;s_2\ge d-ϵ}{e}^{bk{s}_2}ds+O(e^{bk(d-ϵ)}),b\to \infty .$$

$$J:={\int }_{S;s_2\ge d-ϵ}{e}^{bk{s}_2}ds={\int }_{d-ϵ}^d{e}^{bky}\ell (y)dy=e^{bkd}{\int }_0^{ϵ}d\eta e^{-bk\eta }c(d-\eta {)}^{\gamma }.$$

$$\underset{b\to \infty }{\lim }\frac{\ln J}{bk}=d.$$

Full text

Formula for the size of the scatterer is derived explicitly in terms of the scattering amplitude corresponding to this scatterer. By the scatterer either a bounded obstacle $D$ or the support of the compactly supported potential is meant.

1 Introduction

Let $D$ be a bounded $C^{2}-$smooth connected domain in ${\mathbb{R}}^{3}$, $S$ be its boundary, $N$ is the outer unit normal to $S$, $u_{N}$ is the normal derivative of $u$ on $S$, $S^{2}$ be the unit sphere in ${\mathbb{R}}^{3}$, $D^{\prime}:={\mathbb{R}}^{3}\setminus D$, $k=const>0$, $\alpha,\beta\in S^{2}$, $v$ be the scattered field, $g=g(x,y):=\frac{e^{ik|x-y|}}{4\pi|x-y|}$, $\{e_{j}\}_{j=1}^{3}$ be an orthonormal basis in ${\mathbb{R}}^{3}$. We drop the $k-$dependence since $k$ is fixed.

Consider the scattering problem

[TABLE]

where $u_{0}:=e^{ik\alpha\cdot x}$ is the incident field, and $v$, the scattered field, satisfies the radiation condition

[TABLE]

uniformly with respect to $\beta$. Problem (1)–(2) is the obstacle scattering problem, $D$ is the obstacle, the scatterer, $u=u(x):=u(x,\alpha)$ is the scattering solution.

It is known (see, for example, [2], [3]) that problem (1)–(2) has a solution and this solution $u$ is unique.

By the Green’s formula one gets

[TABLE]

Let $r\to\infty,x/r=\beta$ in (3). Then

[TABLE]

where

[TABLE]

The properties of $A(\beta,\alpha)$ are studied, for example, in [2], [3].

It was proved in [2], p.62, that $A(\beta,\alpha)$ is an analytic function of $\beta$ and $\alpha$ in the algebraic variety $M\subset\mathbb{C}^{3}$ defined by the equation $z\cdot z=1$, $z\in\mathbb{C}^{3}$, where $z\cdot z:=\sum_{j=1}^{3}z_{j}^{2}$, $z_{j}\in\mathbb{C}^{1}$. Indeed, from (5) it follows that $A(\beta,\alpha)$ is an analytic function of $\beta$ on the variety $M$. This function is originally defined on $S^{2}$. For small $p>0$ it is defined for $|\beta_{1}|\leq p,|\beta_{2}|\leq p$ and $\beta_{3}=(1-\beta_{1}^{2}-\beta_{2}^{2})^{1/2}$ and is an analytic function of the three complex variables $\beta_{1},\beta_{2},\beta_{3}$ in the region $|\beta_{1}|\leq p,|\beta_{2}|\leq p,\beta_{3}=(1-\beta_{1}^{2}-\beta_{2}^{2})^{1/2}$. The scattering amplitude $A(\beta,\alpha)$ admits a unique analytic continuation to the algebraic variety $M$. Therefore, one can take $\beta$ in formula (5), for example, as

[TABLE]

This $\beta$ clearly belongs to $M$.

Let us call a plane $P$ supporting $D$ at a point $s$ if $s\in D\cap P$ and $D$ is contained in one of the two half-spaces bounded by $P$. A plane is supporting $D$ if it is supporting $D$ at some point. The distance $d$ between two parallel planes supporting $D$ we call the size of $D$ in the direction of the normal to these planes.

Our first result is the following theorem.

Theorem 1. The size of $D$ in the direction $e_{2}$ is

[TABLE]

*where $a^{2}-b^{2}=1$. *

Consider now the potential scattering:

[TABLE]

where $\psi$ satisfies the radiation condition (2), the potential $q$ is a real-valued compactly supported function with support $D$, $q\in L^{2}(D)$. It is known that $\psi$ satisfies relation (4) with

[TABLE]

The function $H$ does not depend on $\beta$. One has

[TABLE]

where $c>0$ is a constant.

Since $S$ is $C^{2}-$smooth, any section of $S$ by a plane is a curve with a bounded length.

Our second result is:

Theorem 2. Formula (7) holds for the estimate of the support $D$ of $q$ in the direction $e_{2}$.

In the next section proofs are given.

2 Proofs

Proof of Theorem 1. Using formula (5) with $\beta$ from (6), one gets

[TABLE]

The conclusion of Theorem 1 follows from (11) and Lemma 1.

Lemma 1. One has

[TABLE]

where $d$ is the size of $D$ in the direction $e_{2}$.

Theorem 1 is proved. $\Box$

Proof of Lemma 1. One has

[TABLE]

We assume that $S$ is $C^{2}-$smooth. Therefore the section of $S$ by any plane $y=const$ is a curve that has has a finite length $\ell(y)$, $\ell(y)=cy^{\gamma}$, where $\gamma>0$ depends on the smoothness of $S$ at the supporting point. In the coordinate system that we use one of the planes, supporting $D$, has the equation $y=0$ and the other plane, supporting $D$, is $y=d$. If one changes variable $y$ setting $d-y=\eta$, then

[TABLE]

Thus

[TABLE]

Lemma 1 is proved. $\Box$

Proof of Theorem 2 is essentially the same as the proof of Theorem 1.

3 Remarks

1. We have chosen $\beta$ by formula (6), but we can choose it so that Im$\beta$ is directed along a given unit vector from $S^{2}$.

2. The symmetry of the scatterer can influence the functional form of the scattering amplitude.

In [4] it is proved that if and only if the scatterer is spherically symmetric its scattering amplitude is a function of $\beta\cdot\alpha$, $A=A(\beta\cdot\alpha)$. Therefore, when the scatterer is spherically symmetric $\alpha$ should not be chosen orthogonal to the vector that is parallel to Im$\beta$.

3. In quantum mechanics the scattering amplitude corresponding to a spherically symmetric compactly supported potential is calculated as a series [1]. This series is less convenient for passing to the limit $b\to\infty$ than formula (9).