ASYMPTOTICS OF THE EIGENVALUES OF A PERIODIC BOUNDARY VALUE PROBLEM FOR A DIFFERENTIAL OPERATOR OF ODD ORDER WITH SUMMABLE OPERATOR

Abstract

The paper is devoted to the study of spectral properties of differential operators of arbitrary odd order with a summable potential and periodic boundary conditions. For large values of the spectral parameter the asymptotics of the solutions of the differential equation that defines the differential operator is obtained. The differential equation that defines the differential operator is reduced to the Volterra integral equation. The integral equation is solved by Picard's method of successive approximations. The method of studying of operators with a summable potential is an extension of the method of studying operators with piecewise smooth coefficients. The study of periodic boundary conditions leads to the study of the roots of the entire function represented in the form of an arbitrary odd-order determinant. To obtain the roots of this function, the indicator diagram has been examined. The roots of this equation are in the sectors of an infinitesimal angle, determined by the indicator diagram. In the paper the asymptotics of eigenvalues of the differential operator under consideration is found. The obtained formulas make it impossible to study the spectral properties of the eigenfunctions and to derive the formula for the first regularized trace of the differential operator under study.

Keywords: The differential operator of odd order, spectral parameter, summable potential, periodic boundary conditions, indicator diagram, asymptotics of solutions, asymptotics of eigenvalues.