Inverse scattering, full-line case
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:'
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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 .
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].
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|
Inverse scattering, full-line case. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inverse_scattering,_full-line_case&oldid=16187