Obstacle scattering

Let be a bounded domain with boundary . A large amount of literature, going back to the mid-1930s, deals with wave scattering by obstacles when is smooth, for example, a -surface, . (See [a3], [a17], [a2].)

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

(a1)

(a2)

(a3)

Here ( the Dirichlet condition), or (the Neumann condition) or (the Robin condition), where is the unit normal to pointing into and is a continuous function (cf. also Dirichlet boundary conditions; Neumann boundary conditions). Condition (a3) is the radiation condition, which selects a unique solution to problem (a1)–(a3). In (a3), is a given unit vector, the direction of the incident plane wave , and 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 [a3].

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

The scattering field in (a3) has the following asymptotics:

(a4)

The function is called the scattering amplitude. This function has the following properties [a17]:

i) realness: , , the bar stands for complex conjugation;

ii) reciprocity: ;

iii) unitarity:

and its consequence, which is called the optical theorem:

where is called the cross-section.

The function is analytic with respect to in and meromorphic in ; it is analytic with respect to and on the variety , where , see [a17], [a18] (cf. also Analytic function; Meromorphic function).

Necessary and sufficient conditions for a scatterer to be spherically symmetric is: , where is the dot product [a18], [a8].

The solution to (a1)–(a3) is called the scattering solution. Any can be expanded with respect to scattering solutions:

The operator is unitary: , , see [a17] (cf. also Unitary operator).

The above results hold in with odd . In with even the scattering amplitude as a function of complex has a logarithmic branch point at .

The scattering problem with minimal assumptions on the smoothness of the boundary is studied in [a13]. If , then existence and uniqueness of the scattering solution have been proved without any assumption on the smoothness of the boundary of a bounded domain . In this case a weak formulation of problem (a1)–(a3) is considered and the limit-absorption principle has been proved.

If , then again a weak formulation of (a1)–(a3) is considered and the only assumption on the smoothness of the boundary is compactness of the embedding into , where and is a ball which contains . Existence and uniqueness of the scattering solution have been proved and the limiting-absorption principle has been established. Finally, if , then the same results are obtained under the assumptions of compactness of the embeddings and , where is equipped with the -dimensional Hausdorff measure and is the Sobolev space.

For example, the embedding is compact for C-domains, that is, domains whose boundary can be covered by finitely many sets open in and on each of these sets the equation of in a local coordinate system can be written as , where is a continuous function.

The scattering problem for one obstacle, small in comparison with the wavelength (, where is the diameter of the small obstacle), for many such bodies (the number of bodies of order ), and in a medium consisting of many such bodies randomly distributed in the space, has been studied in [a7] and later in [a16]. Formulas for the scattering amplitude for acoustic and electromagnetic wave scattering by small bodies of arbitrary shapes are derived in [a16].

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

where is the electrical capacitance of the body ;

where over the repeated indices one sums up from to , is the volume of and is the magnetic polarizability tensor;

where is the area of the boundary .

The -matrix for electromagnetic wave scattering by a small homogeneous body of arbitrary shape is:

where and are tensors of magnetic and electric polarizability and is the angle between the direction of the incident field and the direction of the scattered field; is the magnetic permeability of the medium in which the small body is embedded (cf. also Scattering matrix).

Formulas for the tensors and are derived in [a7] and [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 [a16] and [a6].

Scattering by random surfaces is studied in [a1].

Inverse obstacle scattering.

Three inverse obstacle scattering problems are of interest:

ISP1) Given for all and all , being fixed, find and the boundary condition on .

ISP2) Given for all being fixed, find and the boundary condition on .

ISP3) Given for all and being fixed, find .

Uniqueness of the solution to ISP1) (for ) was first proved by M. Schiffer (1964), whose argument is given in [a17].

Uniqueness of the solution to ISP2) was first proved by A.G. Ramm (1985) and his proof is given in [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 [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 for any bounded domain (without any assumptions on the smoothness of its boundary ), is the Sobolev space which is the closure in of . It is known [a5] that 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 into exists, see [a5]).

Therefore the spectrum of a Neumann Laplacian in such a rough domain for which the imbedding is not compact is not discrete. One way out is given in [a12] and another one in [a11].

Suppose that and are scattering amplitudes at a fixed for two obstacles and let .

Assume that the boundaries of the two obstacles are , , that is, in local coordinates these boundaries , , have equations , where , , .

Let denote the Hausdorff distance between and : .

The basic stability result [a10] is:

where and are positive constants.

In [a10] a yet open problem (as of 2000) is formulated: Derive an inversion formula for finding , given the data , .

The existence of such a formula is proved in [a10]: if is the characteristic function of and is its Fourier transform, then there exists a function such that

where , , is an arbitrary vector. A formula for calculating , , given , , is derived in [a10]. The problem is to construct from the data algorithmically. For inverse potential scattering this is done in [a15].

References

[a1]	F. Bass, I. Fuks, "Wave scattering from statistically rough surfaces" , Pergamon (1979)
[a2]	D. Colton, R. Kress, "Integral equations methods in scattering theory" , Wiley (1983)
[a3]	V. Kupradze, "Randwertaufgaben der Schwingungstheorie und Integralgleichungen" , DVW (1956)
[a4]	R. Leis, "Initial boundary value problems in mathematical physics" , New York (1986)
[a5]	V. Maz'ja, "Sobolev spaces" , Springer (1985)
[a6]	V. Marchenko, E. Khruslov, "Boundary value problems in domains with granulated boundary" , Nauk. Dumka, Kiev (1974) (In Russian)
[a7]	A.G. Ramm, "Theory and applications of some new classes of integral equations" , Springer (1980)
[a8]	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
[a9]	A.G. Ramm, "Stability estimates for obstacle scattering" J. Math. Anal. Appl. , 188 : 3 (1994) pp. 743–751
[a10]	A.G. Ramm, "Stability of the solution to inverse obstacle scattering problem" J. Inverse Ill-Posed Probl. , 2 : 3 (1994) pp. 269–275
[a11]	A.G. Ramm, "New method for proving uniqueness theorems for obstacle inverse scattering problems" Appl. Math. Lett. , 6 : 6 (1993) pp. 19–22
[a12]	A.G. Ramm, "Uniqueness theorems for inverse obstacle scattering problems in Lipschitz domains" Applic. Anal. , 59 (1995) pp. 377–383
[a13]	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
[a14]	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
[a15]	A.G. Ramm, "Stability estimates in inverse scattering" Acta Applic. Math. , 28 : 1 (1992) pp. 1–42
[a16]	A.G. Ramm, "Iterative methods for calculating the static fields and wave scattering by small bodies" , Springer (1982)
[a17]	A.G. Ramm, "Scattering by obstacles" , Reidel (1986)
[a18]	A.G. Ramm, "Multidimensional inverse scattering problems" , Longman/Wiley (1992)
[a19]	A.G. Ramm, "Spectral properties of the Schroedinger operator in some domains with infinite boundaries" Soviet Math. Dokl. , 152 (1963) pp. 282–285
[a20]	A.G. Ramm, "Reconstruction of the domain shape from the scattering amplitude" Radiotech. i Electron. , 11 (1965) pp. 2068–2070
[a21]	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
[a22]	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
[a23]	A.G. Ramm, "On inverse diffraction problem" J. Math. Anal. Appl. , 103 (1984) pp. 139–147
[a24]	F. Ursell, "On the exterior problems of acoustics" Proc. Cambridge Philos. Soc. , 74 (1973) pp. 117–125 (See also: 84 (1978), 545-548)
[a25]	I. Vekua, "Metaharmonic functions" Trudy Tbil. Math. Inst. , 12 (1943) pp. 105–174 (In Russian)