Inverse Spectral Problem for the One-Dimensional Stark Operator on the Semiaxis
- PDF / 195,309 Bytes
- 17 Pages / 594 x 792 pts Page_size
- 18 Downloads / 141 Views
INVERSE SPECTRAL PROBLEM FOR THE ONE-DIMENSIONAL STARK OPERATOR ON THE SEMIAXIS A. R. Latifova1 and A. Kh. Khanmamedov2,3
UDC 517.91
d2 We consider the Stark operator T = − 2 + x + q (x) on the semiaxis 0 x < 1 with Dirichlet dx boundary condition at the origin. By the method of transformation operators, we study the direct and inverse spectral problems, deduce the main integral equation for the inverse problem, and prove that this equation is uniquely solvable. We also propose an effective algorithm of reconstruction of the perturbation potential.
1. Introduction The problem of investigation of the Stark operator is of significant interest for the spectral theory. Various aspects of the direct and inverse spectral problems for this operator have been studied by numerous authors (see [1–12] and the references therein). The present paper is a continuation of the work [13], where we have studied the asymptotics of spectrum of the one-dimensional Stark operator on the semiaxis [0, 1) with Dirichlet boundary condition at the origin. Consider a boundary-value problem generated on the semiaxis 0 x < 1 by a differential equation −y 00 + xy + q(x)y = λy,
(1.1)
λ 2 C,
with a boundary condition (1.2)
y(0) = 0 in the case where the function q(x) is real and satisfies the conditions q(x) 2 C
(1)
[0, 1),
q(x) = o(x),
x ! 1,
Z1 0
� 4 � �x q(x)� dx < 1.
(1.3)
In [13], we have proved that, under conditions (1.3), problem (1.1), (1.2) has a pure discrete spectrum formed by prime eigenvalues λn , n = 1, 2, . . . , where λn ! +1 as n ! 1. Moreover, the eigenvalues λn , n = 1, 2, . . . , are the roots of the function f (0, λ), where f (x, λ) is a solution of Eq. (1.1) with the following asymptotics: � � f (x, λ) = Ai(x − λ) 1 + o(1) as x ! 1,
and Ai(z) is the Airy function of the first kind. 1
Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku; Baku State University, Baku, Azerbaijan; Azerbaijan University, Baku, Azerbaijan; e-mail: [email protected]. 2 Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku; Baku State University, Baku, Azerbaijan; Azerbaijan University, Baku, Azerbaijan; e-mail: agil [email protected]. 3 Corresponding author. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 4, pp. 494–508, April, 2020. Original article submitted January 9, 2020. 568
0041-5995/20/7204–0568
© 2020
Springer Science+Business Media, LLC
I NVERSE S PECTRAL P ROBLEM FOR THE O NE -D IMENSIONAL S TARK O PERATOR ON THE S EMIAXIS
569
A collection of quantities {λn , ↵n > 0}1 n=1 , where ↵n =
sZ
0
1�
� �f (x, λn )�2 dx,
is called the spectral data of problem (1.1), (1.2). In the present paper, we study the inverse spectral problem for the boundary-value problem (1.1), (1.2), i.e., the problem of reconstruction of the potential q(x) from class (1.3) according to the available spectral data. Note that the inverse scattering problem for Eq. (1.1) given on the entire axis was studied by the method of transformation operator i
Data Loading...
