Completeness of the set $\{e^{ik\beta \cdot s}\}|_{\forall \beta \in S^2}$

TL;DR

This paper proves that the set of exponential functions with directions on the unit sphere is complete in the space of square-integrable functions on a smooth surface, under certain spectral conditions.

Contribution

It establishes the completeness of a specific exponential set in L^2 space on a surface, under the condition that k^2 is not a Dirichlet eigenvalue.

Findings

The set $\

is total in L^2(S).

Completeness holds for fixed k > 0 when $k^2$ is not a Dirichlet eigenvalue of the Laplacian in D.

Abstract

It is proved that the set $\{e^{ik\beta \cdot s}\}|_{\forall \beta \in S^2}$, where $S^2$ is the unit sphere in $\mathbb{R}^3$, $k>0$ is a fixed constant, $k^2$ is not a Dirichlet eigenvalue of the Laplacian in $D$, $s\in S$, is total in $L^2(S)$. Here $S$ is a smooth, closed, connected surface in $\mathbb{R}^3$.

Equations

$${\int }_{S}f(s)e^{ik\beta \cdot s}ds=0\forall \beta \in S^2,$$

$$({\nabla }^2+k^2)u=0inD,u{|}_S=0.$$

$$w(x):={\int }_{S^2}h(\beta )e^{ik\beta \cdot x}.$$

$$({\nabla }^2+k^2)w=0in{\mathbb{R}}^3.$$

$${\int }_{S}f(s)w(s)ds=0$$

$$({\nabla }^2+k^2)G=({\nabla }^2+k^2)FinD;G{|}_S=0.$$

$$0={\int }_D({\nabla }^2+k^2)Fvdx,$$

$${\int }_S\psi v_{N}ds=0.$$

Full text

It is proved that the set $\{e^{ik\beta\cdot s}\}|_{\forall\beta\in S^{2}}$, where $S^{2}$ is the unit sphere in $\mathbb{R}^{3}$, $k>0$ is a fixed constant, $s\in S$, is total in $L^{2}(S)$ if and only if $k^{2}$ is not a Dirichlet eigenvalue of the Laplacian in $D$. Here $S$ is a smooth, closed, connected surface in $\mathbb{R}^{3}$.

1 Introduction

Let $D\subset\mathbb{R}^{3}$ be a bounded domain with a connected closed $C^{2}-$smooth boundary $S$, $D^{\prime}:=\mathbb{R}^{3}\setminus D$ be the unbounded exterior domain and $S^{2}$ be the unit sphere in $\mathbb{R}^{3}$, $\beta\in S^{2}$, $s\in S$.

We are interested in the following problem:

Is the set $\{e^{ik\beta\cdot s}\}|_{\forall\beta\in S^{2}}$ total in $L^{2}(S)$?

A set $\{\phi(s,\beta)\}$ is total (complete) in $L^{2}(S)$ if the relation $\int_{S}f(s)\phi(s,\beta)ds=0$ for all $\beta\in S^{2}$ implies $f=0$, where $f\in L^{@}(S)$ is an arbitrary fixed function.

The above question is of interest by itself, but also it is of interest in scattering problems and in inverse problems, see [1]–[5].

Our result is:

Theorem 1. The set $\{e^{ik\beta\cdot s}\}|_{\forall\beta\in S^{2}}$ is total in $L^{2}(S)$ if and only if $k^{2}$ is not a Dirichlet eigenvalue of the Laplacian in $D$.

2 Proof of Theorem 1

Necessity. Let $f\in L^{2}(S)$ and

[TABLE]

and there is a $u\not\equiv 0$ such that

[TABLE]

Choose $f=u_{N}$, where $N$ is the unit normal to $S$ pointing out of $D$. Then, by Green’s formula, equation (1) holds and $f\not\equiv 0$ by the uniqueness of the solution to the Cauchy problem for elliptic equation (2). Necessity is proved.

Sufficiency. Assume that $f\in L^{2}(S)$ is and arbitrary fixed function, $f\not\equiv 0$, and (1) holds. Let $h\in L^{2}(S^{2})$ be arbitrary and

[TABLE]

Then

[TABLE]

If (1) holds, then

[TABLE]

for all $w$ of the form (3). Let us now apply the following Lemma:

Lemma 1. The set $\{w|_{S}\}$ for all $h\in L^{2}(S^{2})$ is the orthogonal complement in $L^{2}(S)$ to the linear span of the set $\{v_{N}\}$, where $v$ solve equation (4) and $v|_{S}=0$.

If $k^{2}$ is not a Dirichlet eigenvalue of the Laplacian in $D$, then Lemma 1 implies that the set $\{w|_{S}\}$ is total in $L^{2}(S)$, so (1) implies $f=0$. Sufficiency and Theorem 1 are proved. $\Box$

Lemma 1 is similar to Theorem 6 in [3].

Proof of Lemma 1. Let $w|_{S}:=\psi$. Choose an arbitrary $F\in C^{2}(D)$ such that $F|_{S}=\psi$. Define $G:=F-w$ in $D$. Then

[TABLE]

For (6) to hold it is necessary and sufficient that

[TABLE]

where $v$ is an arbitrary function in the set of solutions of equation (2). Using Green’s formula one reduces condition (7) to the following condition:

[TABLE]

Therefore the set $\{\psi\}$ is the orthogonal complement in $L^{2}(S)$ of the linear span of the functions $\{v_{N}\}$. Lemma 1 is proved. $\Box$