Inverse scattering, multi-dimensional case

There are many multi-dimensional inverse scattering problems. Below, inverse potential scattering and inverse geophysical scattering are briefly discussed; see Obstacle scattering for inverse obstacle scattering problems.

Inverse potential scattering.

To formulate the inverse potential scattering problem, consider first the direct scattering problem (see [a1], [a2], [a4], [a5], [a6], Appendix):

(a1)

(a2)

(a3)

where is given, is the unit sphere, is the scattered field, is the scattering solution, condition (a3) is called the (outgoing) radiation condition, is the incident plane wave, and is a real-valued function, called a potential,

The existence and uniqueness of the solution to (a1)–(a3) has been proved under less restrictive assumptions on [a2]. The function has the form

where the coefficient is called the scattering amplitude.

The inverse potential scattering problem consists of finding given on some subsets of .

The first result is simple: If is known for all and all , then is uniquely determined.

If

then it is known (e.g. [a6], p. 233, see also [a4]) that

so that can be found:

The second result is much more difficult.

For decades it was not known if the data , , and fixed, determine uniquely. In 1987 the uniqueness result has been established by A.G. Ramm (see [a7], [a8]) under the assumptions , for , where is an arbitrary large fixed number, and in 1988 inversion procedures were published; see [a7]. One of them, proposed by Ramm, is based on the formula

where , , , and is an arbitrary point.

Another inversion procedure ([a3], [a7]) is based on the reconstruction of the Dirichlet-to-Neumann mapping and then finding .

Error estimates for Ramm's inversion procedure in the case of noisy data and an algorithm for calculating the function in the inversion formula are obtained in [a9].

The uniqueness problem for inverse potential scattering with the data , , , , fixed, is still open (as of 2000).

The same is true for the uniqueness problem for inverse potential scattering with the (backscattering) data , , , although for this problem a uniqueness theorem for small holds.

Inverse geophysical scattering.

The inverse geophysical scattering problem consists of finding the unknown coefficient in the equation

(a4)

where satisfies the outgoing radiation condition (a3), is fixed, and is a real-valued function with compact support in .

The scattering data are the values , , that is, the values of on the surface of the Earth. The function describes an inhomogeneity in the velocity profile (in the refraction coefficient), can be an acoustic pressure. Uniqueness of the solution to inverse geophysical scattering problem was proved in 1987 [a8], [a7].

The uniqueness problem for inverse geophysical scattering with data , , , and fixed, is open (as of 2000).

A reduction of the inverse geophysical scattering problem with the data , , to the inverse potential scattering problem with the data , , fixed, , with the unit vector along -axis, is done in [a7].

Inverse potential scattering: Open problem.

An interesting open problem (as of 2000) in inverse potential scattering is the problem of finding discontinuities of and the number of bound states of the Schrödinger operator generated by the expression in from the knowledge of fixed energy scattering data , .

If , then is an analytic function of . Therefore, knowledge of on an open set in , however small, allows one to recover on .

The assumption concerning compactness of the support of is natural in inverse potential scattering because the scattering data are always noisy and it is not possible in principle to recover the tail of a (that is, for , where is sufficiently large) from knowledge of noisy data ,

(see [a7] for a proof).

References

[a1]	H. Cycon, R. Froese, W. Kirsch, B. Simon, "Schrödinger operators" , Springer (1986)
[a2]	L. Hörmander, "Analysis of linear partial differential operators" , IV , Springer (1985)
[a3]	A. Nachman, "Reconstruction from boundary measurements" Ann. Math. , 128 (1988) pp. 531–578
[a4]	R. Newton, "Inverse Schrödinger scattering in three dimensions" , Springer (1989)
[a5]	D. Pearson, "Quantum scattering and spectral theory" , Acad. Press (1988)
[a6]	A.G. Ramm, "Random fields estimation theory" , Longman/Wiley (1990)
[a7]	A.G. Ramm, "Multidimensional inverse scattering problems" , Longman/Wiley (1992)
[a8]	A.G. Ramm, "Recovery of the potential from fixed energy scattering data" Inverse Probl. , 4 (1988) pp. 877–886 (See also: Ibid. 3 (1987), L77-82)
[a9]	A.G. Ramm, "Stability estimates in inverse scattering" Acta Applic. Math. , 28 : 1 (1992) pp. 1–42
[a10]	A.G. Ramm, "Stability of solutions to inverse scattering problems with fixed-energy data" Rend. Sem. Mat. e Fisico (2001) pp. 135–211