Inverse scattering, full-line case

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

(a1)

(a2)

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 :

(a3)

(a4)

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:

Then

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

where

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 (a5) 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].

References

[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