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>
|
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 |
Inverse scattering, full-line case. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inverse_scattering,_full-line_case&oldid=43394