On the asymptotic behavior of the spectrum of a sixth-order differential operator, whose potential is the delta function

Abstract

In this paper we propose a new method for studying differential operators with discontinuous coefficients.We consider a sequence of sixth-order differential operators with piecewise-smooth coefficients. The limit of the sequence of these operators’ potentials is the Dirac delta function. The boundary conditions are separated. To correctly determine solutions of differential equations with discontinuous coefficients at the points of discontinuity, “gluing” conditions are required. Asymptotic solutions were written out for large values of the spectral parameter, with the help of them the “gluing” conditions were studied and the boundary conditions were investigated. As a result, we derive an eigenvalues equation for the operator under study, which is an entire function. The indicator diagram of the eigenvalues equation, which is a regular hexagon, is investigated. In various sectors of the indicator diagram, the method of successive approximations has been used to find the eigenvalues asymptotics of the studied differential operators. The limit of the asymptotic of the spectrum determines the spectrum of the sixth-order operator, whose potential is the delta function.