﻿ Inverse Scattering Problem for the Schrödinger’s Equation

International Journal of Modern Nonlinear Theory and Application
Vol.2 No.4(2013), Article ID:41231,3 pages DOI:10.4236/ijmnta.2013.24035

Inverse Scattering Problem for the Schrödinger’s Equation

Asset A. Durmagambetov, Leyla S. Fazilova

Buketov Karaganda State University, Karaganda, Kazakhstan

Email: aset.durmagambet@gmail.com

Copyright © 2013 Asset A. Durmagambetov, Leyla S. Fazilova. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received November 11, 2013; revised December 11, 2013; accepted December 18, 2013

Keywords: Schrödinger’s Equation; Potential; Scattering Amplitude

ABSTRACT

The analytic properties of the scattering amplitude are discussed. And, the representation of the potential by the scattering amplitude is obtained.

1. Introduction

We consider that the operators, are defined in the dense set in the space and that is a bounded fast-decreasing function. The operator is called Schrödinger’s operator.

We consider the three-dimensional inverse scattering problem for the Schrödinger’s operator: the scattering potential has to be reconstructed from scattering amplitude. This problem has been studied by a number of researchers (in [1-3] and references therein).

2. Results

We consider Schrödinger’s equation:

(1)

Let is a solution of the (1) with the following asympotic behavior:

(2)

where scattering amplitude, and

, for

(3)

We also define the solution, for

as.

As well known [1]:

(4)

This equation is the key to solving the inverse scattering problem, and was first used by R. G. Newton in [2,3] and E. Somersalo et al. in [4].

Equation (4) is equivalent to the following:

(5)

where is a scattering operator with the kernel,

Here is a theorem according to [1]:

Theorem 1 (The energy and momentum conservation laws)

Let, then where is a unitary operator.

Definition 1 The set of measurable functions with the norm, defined as

is recognized as Rollnik’ class.

As shown in [5], is an orthonormal system of eigenfunctions for the continuous spectrum. In addition to the continuous spectrum there are a finite number of negative eigenvalues, designated as with corresponding normalized eigenfunctions, where.

We present Povzner’s results [5] below:

Theorem 2 (Completness) Both for arbitrary and for eigenfunctions Parseval’s identity is valid.

(6)

where and are Fourier coefficients for continuous and dicrete cases.

Theorem 3 (Birmann-Schwinger estimation). Let. Then number of discrete eigenvalues can be estimated as:

(7)

The theorem was proved in [6].

Let introduce the following notation:

(8)

(9)

where. Define the operators

, for as follows:

(10)

(11)

(12)

We consider the Riemann problem of finding a function, which is analytic in the complex plane with cut along the real axis. values on the sides of the cat are denoted as,. Below present the results of [7].

Lemma 1

(13)

Theorem 4 Let,. Then

(14)

The proof of the above follows from the classic results on the Riemann problem.

Lemma 2 Let , then

(15)

The proof of the above follows from the definitions of functions.

Lemma 3 Let,

then

(16)

The proof of the above follows from the definitions of functions.

Lemma 4 Let, then

(17)

The proof of the above follows from the definitions of functions and Theorem 1.

Definition 2 Denote by the set of functions with the norm

Definition 3 Denote by the set of functions g such that

.

Lemma 5 Supposethen the operator defined on the set has inverse defined on the.

The proof of the above follows from the definitions of and conditions Lemma 5

Lemma 6 Let and is existing. Then

(18)

(19)

(20)

The proof of the above follows from the definitions of functions and Equation (4)

Lemma 7 Let. Then

(21)

The lemma can be proved substituting in Equation (1).

Lemma 8 Let, and is existing. Then

(22)

(23)

The proof of the above follows from the definitions of, Lemma 6, Lemma 7.

3. Conclusion

This study has shown once again the outstanding properties of the scattering operator, which allow, in combination with analytical properties of the wave function, obtaining the almost explicit formulas for the potential from the scattering amplitude. Furthermore, this approach allows solving the problem of over-determination, resulting from the fact that the potential is a function of three variables, whereas the amplitude is a function of five variables. We have shown that it is sufficient to average the scattering amplitude to eliminate the two extra variables.

4. Acknowledgements

We thank the Ministry of Education and Science of the Republic of Kazakhstan for a grant, and the “Factor” Company of System Researches for combining our efforts in this project.

The work was performed as a part of the international project “Joint Kazakh-Indian study the influence of anthropogenic factors on atmospheric phenomena on the basis of numerical weather prediction models WRF (Weather Research and Forecasting)”, commissioned by the Ministry of Education and Science of the Republic of Kazakhstan.

、References

[1]       L. D Faddeev, “Inverse Problem of Quantum Scattering Theory II,” Itogi Nauk Takh. Sov Probl Mat, Vol. 3, 1974, p. 93-180.

[2]       R. G. Newton, “New Result on the Inverse Scattering Problem in Three Dimensions,” Physical Review Letters, Vol. 43, No. 8, 1979, pp. 541-542.

[3]       R. G. Newton, “Inverse Scattering Three Dimensions,” Journal of Mathematical Physics, Vol. 21, No. 7, 1980, pp. 1698-1715. http://dx.doi.org/10.1063/1.524637

[4]       E. Somersalo, et al., “Inverse Scattering Problem for the Schrodinger’s Equation in Three Dimensions: Connections between Exact and Approximate Methods,” 1988.

[5]       A. Y. Povzner, “On the Expansion of Arbitrary Functions in Characteristic Functions of the Operator ,” Matematicheskii Sbornik, Vol. 32, No. 74, 1953, pp. 109- 156.

[6]       M. Birman, “On the Spectrum of Singular BoundaryValue Problems,” (Russian) Matematicheskii Sbornik, Vol. 55, No. 97, 1961, pp. 125-174.

[7]       H. Poincare, “Lecons de Mecanique Celeste,” 1910.