On the asymptotic behavior of the eigenvalues of differential operator of even order with discontinuous weight function

S. I. Mitrokhin

Lomonosov Moscow State University, Research Computing Center

Abstract: We study a high-order differential operator with discontinuous coefficients. The potential of the operator is assumed to be a piecewise-smooth function on a finite segment of the operator specification, the weight function is piecewise constant. At the point of discontinuity of the weight function, the conditions of "conjugation" are required. Separated boundary conditions of a general form are studied. The asymptotic behavior of the fundamental system of solutions of differential equations that define the operator under study is studied. With the help of this asymptotics, the conditions of "conjugation" of the investigated differential operator are studied. Then the boundary conditions of the operator under study are investigated. The spectral properties of the differential operator, which arises in practice in the study vibrations of beams, bridges, and membranes composed of materials of different densities, are studied. An equation for the eigenvalues of the operator is derived, which is an entire function, for finding the roots of which an indicator diagram is studied. It is proved that the indicator diagram is a regular polygon. In various sectors of the indicator diagram, the asymptotics of the eigenvalues of the differential operator under study is found. In the case of passing to the limit, the resulting formula leads to a formula for a classical operator with a smooth potential and a constant weight function. Using the found formulas, it is possible to calculate the first regularized trace of the operator under study.

Keywords: spectral parameter, differential operator, weight function, asymptotics of solutions, the spectrum of operator, indicator diagram.

UDC: 517.47

MSC: 00000