Inverse scattering, full-line case

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Let , where the bar stands for complex conjugation. Consider the (direct) scattering problem:


The coefficients and are called the reflection and transmission coefficients. One can prove that is analytic in except at a finite number of points , , , which are simple poles of .

Problem (a1)–(a2) describes scattering by a plane wave falling from and scattered by the potential .

One can also consider the scattering of the plane wave falling from :


One proves that , , , where the bar stands for complex conjugation, . The matrix

is called the -matrix (cf. Scattering matrix). Conservation of energy implies .

Let and be the solutions to (a1) satisfying the conditions:


where are the kernels which define the transformation operators. One has


The function is analytic in and has finitely many simple zeros all of which are at the points , , , , .

If , then ,

The numbers are the eigenvalues of the operator in . They are called the bound states.

The scattering data are the values

The inverse scattering problem (ISP) consists of finding from .

The inverse scattering problem has at most one solution in the class . This solution can be calculated by the following Marchenko method:'

<tbody> </tbody>
1 Define

and solve the following Marchenko equatio for :

If the data correspond to a , then equation (a5) is uniquely solvable in for every .
2 If is found, then .

The main result [a7] is the characterization property for the scattering data: In order that be the scattering data corresponding to a , it is necessary and sufficient that the following conditions hold:

i) for , the function for is continuous,

where , and as .

ii) The function

is absolutely continuous and

for every .

iii) Denote

The function is continuous in and

iv) The function

is absolutely continuous and

for every .

A similar result holds for the data

and the potential can be obtained by the Marchenko method, .

In [a2] the above theory is generalized to the case when tends to a different constants as and .

In [a5] a different approach to solving the inverse scattering problem is described for

The approach in [a5] is based on a trace formula.

If for , then the reflection coefficient alone, without the knowledge of and , determines uniquely. A simple proof of this and similar statements, based on property for ordinary differential equations (cf. Ordinary differential equations, property for), is given in [a10].

An inverse scattering problem for an inhomogeneous Schrödinger equation is studied in [a5].

The inverse scattering method is a tool for solving many evolution equations (cf. also Evolution equation) and is used in, e.g., soliton theory [a7], [a1], [a3], [a6] (cf. also Korteweg–de Vries equation; Harry Dym equation).

Methods for adding and removing bound states are described in [a5]. They are based on the Darboux–Crum transformations and commutation formulas.

A large bibliography can be found in [a4].


[a1] M. Ablowitz, H. Segur, "Solutions and inverse scattering transform" , SIAM (1981)
[a2] A. Cohen, T. Kappeler, "Scattering and inverse scattering for step-like potentials in the Schrödinger equation" Indiana Math. J. , 34 (1985) pp. 127–180
[a3] F. Calogero, A. Degasperis, "Solutions and the spectral transform" , North-Holland (1982)
[a4] K. Chadan, P. Sabatier, "Inverse problems in quantum scattering" , Springer (1989)
[a5] P. Deift, E. Trubowitz, "Inverse scattering on the line" Commun. Pure Appl. Math. , 32 (1979) pp. 121–251
[a6] L. Faddeev, L. Takhtadjian, "Hamiltonian methods in the theory of solutions" , Springer (1986)
[a7] V. Marchenko, "Sturm–Liouville operators and applications" , Birkhäuser (1986)
[a8] A.G. Ramm, "Multidimensional inverse scattering problems" , Longman/Wiley (1992)
[a9] A.G. Ramm, "Inverse problem for an inhomogeneous Schrödinger equation" J. Math. Phys. , 40 : 8 (1999) pp. 3876–3880
[a10] A.G. Ramm, "Property C for ODE and applications to inverse problems" A.G. Ramm (ed.) P.N. Shivakumar (ed.) A.V. Strauss (ed.) , Operator Theory and Applications , Fields Inst. Commun. , 25 , Amer. Math. Soc. (2000) pp. 15–75
How to Cite This Entry:
Inverse scattering, full-line case. Encyclopedia of Mathematics. URL:,_full-line_case&oldid=16187
This article was adapted from an original article by A.G. Ramm (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article