Обратная задача на полуоси оператора Штурма-Лиувилля с бесконечнозонным потенциалом

Аннотация

Subjects of the inquiry: The work is devoted to studing to inverse problem by S'-function of scattering theory and by the spectral function for Dirac’s operator. Aim of the inquiry: To study the inverse spectral problem for Dirac’s operator. Method of the inquiry: In the work are used the methods of functional analysis, spectral theory of linear operators, the theory of functions of complex variable and theory differential equations. The results achieved and their novelty: The main results of this work are new and consist of the following: 1) the regularized trace formulas for Dirac’s operator with peculiarity in the potential is calculated by Lax’s method; 2) the inverse scattering problem for Dirac’s operator is studied by following conditions: (a) p(x), q(x) sufficiently quickly tends to zero as x—>oo, where xe[a,oo), a e R'; (b) spectrum of the problem Dy = Лу, yf(a) = 0 (-oo<x<a) is discrete; 3) Dirac operator with mass which has a countable number of eigenvalues in the continuous spectrum is constructed and for the constructed operator the sufficient conditions to belong coefficients p(x), <?(x) to the space Z/’(0,=o) are found; 4) the position’s dependence of eigenvalues in the continuous spectrum Dirac’s operator with mass from the common boundary condition is studied. Practical value: the work have theoretical character. Degree of embed and economic effectivity: on the basis of the received results a special course will be read for the students of the magistracy. Sphere of usage: the obtained results may be used in spectral theory of linear operators, in mathematical physics for integration of nonlinear equation as well as quantum mechanics and other areas of natural subjects.