View source for Obstacle scattering

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o1300101.png" /> be a bounded domain with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o1300102.png" />. A large amount of literature, going back to the mid-1930s, deals with wave scattering by obstacles when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o1300103.png" /> is smooth, for example, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o1300104.png" />-surface, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o1300105.png" />. (See [[#References|[a3]]], [[#References|[a17]]], [[#References|[a2]]].)

The obstacle scattering problem consists of finding the solution to the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o1300106.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o1300107.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o1300108.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o1300109.png" /> ( the Dirichlet condition), or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001010.png" /> (the Neumann condition) or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001011.png" /> (the Robin condition), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001012.png" /> is the unit normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001013.png" /> pointing into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001015.png" /> is a [[Continuous function|continuous function]] (cf. also [[Dirichlet boundary conditions|Dirichlet boundary conditions]]; [[Neumann boundary conditions|Neumann boundary conditions]]). Condition (a3) is the radiation condition, which selects a unique solution to problem (a1)–(a3). In (a3), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001016.png" /> is a given unit vector, the direction of the incident plane wave <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001017.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001018.png" /> is the wave number.

The scattering problem (a1)–(a3) has a solution and the solution is unique. This basic result was proved originally by the integral equations method [[#References|[a3]]].

There are many different types of integral equations which allow one to study problem (a1)–(a3) (see [[#References|[a17]]], where most of these equations are derived).

The scattering field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001019.png" /> in (a3) has the following asymptotics:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001021.png" /></td> </tr></table>

The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001022.png" /> is called the scattering amplitude. This function has the following properties [[#References|[a17]]]:

i) realness: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001024.png" />, the bar stands for complex conjugation;

ii) reciprocity: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001025.png" />;

iii) unitarity:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001026.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001027.png" /></td> </tr></table>

and its consequence, which is called the optical theorem:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001028.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001029.png" /> is called the cross-section.

The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001030.png" /> is analytic with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001032.png" /> and meromorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001033.png" />; it is analytic with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001035.png" /> on the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001037.png" />, see [[#References|[a17]]], [[#References|[a18]]] (cf. also [[Analytic function|Analytic function]]; [[Meromorphic function|Meromorphic function]]).

Necessary and sufficient conditions for a scatterer to be spherically symmetric is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001039.png" /> is the dot product [[#References|[a18]]], [[#References|[a8]]].

The solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001040.png" /> to (a1)–(a3) is called the scattering solution. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001041.png" /> can be expanded with respect to scattering solutions:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001042.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001043.png" /></td> </tr></table>

The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001044.png" /> is unitary: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001046.png" />, see [[#References|[a17]]] (cf. also [[Unitary operator|Unitary operator]]).

The above results hold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001047.png" /> with odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001048.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001049.png" /> with even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001050.png" /> the scattering amplitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001051.png" /> as a function of complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001052.png" /> has a [[Logarithmic branch point|logarithmic branch point]] at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001053.png" />.

The scattering problem with minimal assumptions on the smoothness of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001054.png" /> is studied in [[#References|[a13]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001055.png" />, then existence and uniqueness of the scattering solution have been proved without any assumption on the smoothness of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001056.png" /> of a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001057.png" />. In this case a weak formulation of problem (a1)–(a3) is considered and the [[Limit-absorption principle|limit-absorption principle]] has been proved.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001058.png" />, then again a weak formulation of (a1)–(a3) is considered and the only assumption on the smoothness of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001059.png" /> is compactness of the embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001060.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001061.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001063.png" /> is a ball which contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001064.png" />. Existence and uniqueness of the scattering solution have been proved and the limiting-absorption principle has been established. Finally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001065.png" />, then the same results are obtained under the assumptions of compactness of the embeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001068.png" /> is equipped with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001069.png" />-dimensional [[Hausdorff measure|Hausdorff measure]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001070.png" /> is the Sobolev space.

For example, the embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001071.png" /> is compact for C-domains, that is, domains whose boundary can be covered by finitely many sets open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001072.png" /> and on each of these sets the equation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001073.png" /> in a local coordinate system can be written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001074.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001075.png" /> is a [[Continuous function|continuous function]].

The scattering problem for one obstacle, small in comparison with the wavelength (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001076.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001077.png" /> is the diameter of the small obstacle), for many such bodies (the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001078.png" /> of bodies of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001079.png" />), and in a medium consisting of many such bodies randomly distributed in the space, has been studied in [[#References|[a7]]] and later in [[#References|[a16]]]. Formulas for the scattering amplitude for acoustic and electromagnetic wave scattering by small bodies of arbitrary shapes are derived in [[#References|[a16]]].

These formulas for acoustic wave scattering on a single small body, containing the origin, are

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001080.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001081.png" /> is the electrical capacitance of the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001082.png" />;

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001083.png" /></td> </tr></table>

where over the repeated indices one sums up from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001084.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001086.png" /> is the volume of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001088.png" /> is the magnetic polarizability tensor;

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001089.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001090.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001091.png" /> is the area of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001092.png" />.

The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001094.png" />-matrix for electromagnetic wave scattering by a small homogeneous body of arbitrary shape is:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001095.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001096.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001097.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o13001099.png" /> are tensors of magnetic and electric polarizability and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010100.png" /> is the angle between the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010101.png" /> of the incident field and the direction of the scattered field; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010102.png" /> is the magnetic permeability of the medium in which the small body is embedded (cf. also [[Scattering matrix|Scattering matrix]]).

Formulas for the tensors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010104.png" /> are derived in [[#References|[a7]]] and [[#References|[a16]]].

One can derive an equation for the average field in the medium which consists of many small obstacles (particles) randomly distributed in a region. This done in [[#References|[a16]]] and [[#References|[a6]]].

Scattering by random surfaces is studied in [[#References|[a1]]].

==Inverse obstacle scattering.==
Three inverse obstacle scattering problems are of interest:

ISP1) Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010105.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010106.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010108.png" /> being fixed, find <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010109.png" /> and the boundary condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010110.png" />.

ISP2) Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010111.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010112.png" /> being fixed, find <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010113.png" /> and the boundary condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010114.png" />.

ISP3) Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010115.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010117.png" /> being fixed, find <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010118.png" />.

Uniqueness of the solution to ISP1) (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010119.png" />) was first proved by M. Schiffer (1964), whose argument is given in [[#References|[a17]]].

Uniqueness of the solution to ISP2) was first proved by A.G. Ramm (1985) and his proof is given in [[#References|[a17]]].

A uniqueness theorem for ISP3) has not yet (2000) been proved: it is an open problem to prove (or disprove the existence of) such a theorem.

One can consider inverse obstacle scattering for penetrable obstacles [[#References|[a14]]].

Schiffer's proof of the uniqueness theorem is based on a result saying that the spectrum of the Dirichlet Laplacian in any bounded domain is discrete. This result follows from the compactness of the embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010120.png" /> for any bounded domain (without any assumptions on the smoothness of its boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010121.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010122.png" /> is the [[Sobolev space|Sobolev space]] which is the closure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010123.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010124.png" />. It is known [[#References|[a5]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010125.png" /> is not compact for rough domains (it is compact for Lipschitz domains, for domains satisfying the cone condition, for C-domains, and for E-domains, i.e. domains for which a bounded extension operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010126.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010127.png" /> exists, see [[#References|[a5]]]).

Therefore the spectrum of a Neumann Laplacian in such a rough domain for which the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010128.png" /> is not compact is not discrete. One way out is given in [[#References|[a12]]] and another one in [[#References|[a11]]].

Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010129.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010130.png" /> are scattering amplitudes at a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010131.png" /> for two obstacles and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010132.png" />.

Assume that the boundaries of the two obstacles are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010134.png" />, that is, in local coordinates these boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010136.png" />, have equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010137.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010138.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010139.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010140.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010141.png" /> denote the Hausdorff distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010143.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010144.png" />.

The basic stability result [[#References|[a10]]] is:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010145.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010146.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010147.png" /> are positive constants.

In [[#References|[a10]]] a yet open problem (as of 2000) is formulated: Derive an inversion formula for finding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010148.png" />, given the data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010149.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010150.png" />.

The existence of such a formula is proved in [[#References|[a10]]]: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010151.png" /> is the characteristic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010152.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010153.png" /> is its [[Fourier transform|Fourier transform]], then there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010154.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010155.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010156.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010157.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010158.png" /> is an arbitrary vector. A formula for calculating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010159.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010160.png" />, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010161.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010162.png" />, is derived in [[#References|[a10]]]. The problem is to construct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010163.png" /> from the data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130010/o130010164.png" /> algorithmically. For inverse potential scattering this is done in [[#References|[a15]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Bass,   I. Fuks,   "Wave scattering from statistically rough surfaces" , Pergamon  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Colton,   R. Kress,   "Integral equations methods in scattering theory" , Wiley  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V. Kupradze,   "Randwertaufgaben der Schwingungstheorie und Integralgleichungen" , DVW  (1956)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Leis,   "Initial boundary value problems in mathematical physics" , New York  (1986)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  V. Maz'ja,   "Sobolev spaces" , Springer  (1985)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  V. Marchenko,   E. Khruslov,   "Boundary value problems in domains with granulated boundary" , Nauk. Dumka, Kiev  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A.G. Ramm,   "Theory and applications of some new classes of integral equations" , Springer  (1980)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A.G. Ramm,   "Necessary and sufficient condition for a scattering amplitude to correspond to a spherically symmetric scatterer"  ''Appl. Math. Lett.'' , '''2'''  (1989)  pp. 263–265</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A.G. Ramm,   "Stability estimates for obstacle scattering"  ''J. Math. Anal. Appl.'' , '''188''' :  3  (1994)  pp. 743–751</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  A.G. Ramm,   "Stability of the solution to inverse obstacle scattering problem"  ''J. Inverse Ill-Posed Probl.'' , '''2''' :  3  (1994)  pp. 269–275</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A.G. Ramm,   "New method for proving uniqueness theorems for obstacle inverse scattering problems"  ''Appl. Math. Lett.'' , '''6''' :  6  (1993)  pp. 19–22</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  A.G. Ramm,   "Uniqueness theorems for inverse obstacle scattering problems in Lipschitz domains"  ''Applic. Anal.'' , '''59'''  (1995)  pp. 377–383</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  A.G. Ramm,   M. Sammartino,   "Existence and uniqueness of the scattering solutions in the exterior of rough domains"  A.G. Ramm (ed.)  P.N. Shivakumar (ed.)  A.V. Strauss (ed.) , ''Operator Theory and Its Applications'' , ''Fields Inst. Commun.'' , '''25''' , Amer. Math. Soc.  (2000)  pp. 457–472</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  A.G. Ramm,   P. Pang,   G. Yan,   "A uniqueness result for the inverse transmission problem"  ''Internat. J. Appl. Math.'' , '''2''' :  5  (2000)  pp. 625–634</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  A.G. Ramm,   "Stability estimates in inverse scattering"  ''Acta Applic. Math.'' , '''28''' :  1  (1992)  pp. 1–42</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  A.G. Ramm,   "Iterative methods for calculating the static fields and wave scattering by small bodies" , Springer  (1982)</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  A.G. Ramm,   "Scattering by obstacles" , Reidel  (1986)</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  A.G. Ramm,   "Multidimensional inverse scattering problems" , Longman/Wiley  (1992)</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  A.G. Ramm,   "Spectral properties of the Schroedinger operator in some domains with infinite boundaries"  ''Soviet Math. Dokl.'' , '''152'''  (1963)  pp. 282–285</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  A.G. Ramm,   "Reconstruction of the domain shape from the scattering amplitude"  ''Radiotech. i Electron.'' , '''11'''  (1965)  pp. 2068–2070</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  A.G. Ramm,   "Approximate formulas for tensor polarizability and capacitance of bodies of arbitrary shape and its applications"  ''Soviet Phys. Dokl.'' , '''195'''  (1970)  pp. 1303–1306</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  A.G. Ramm,   "Electromagnetic wave scattering by small bodies of an arbitrary shape"  V. Varadan (ed.) , ''Acoustic, Electromagnetic and Elastic Scattering: Focus on T-Matrix Approach'' , Pergamon  (1980)  pp. 537–546</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top">  A.G. Ramm,   "On inverse diffraction problem"  ''J. Math. Anal. Appl.'' , '''103'''  (1984)  pp. 139–147</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top">  F. Ursell,   "On the exterior problems of acoustics"  ''Proc. Cambridge Philos. Soc.'' , '''74'''  (1973)  pp. 117–125  (See also: 84 (1978), 545-548)</TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top">  I. Vekua,   "Metaharmonic functions"  ''Trudy Tbil. Math. Inst.'' , '''12'''  (1943)  pp. 105–174  (In Russian)</TD></TR></table>