Study of differential operator with summable potential and discontinuous weight function

In the work we propose a new approach for studying differential operators with a discontinuous weight function. We study the spectral properties of a differential operator on a finite segment with separated boundary conditions and with “matching” condition at the discontinuity point of the weight function. We assume that the potential of the operator is a summable function on the segment, on which the operator is considered. For large value of the spectral parameter we obtain an asymptotics for the fundamental system of solutions of the corresponding differential equation. By means of this asymptotics we study the “matching” conditions of the considered differential operator. Then we study the boundary conditions of the considered operator. As a result, we obtain an equation for the eigenvalues of the operator, which an entire function. We study the indicator diagram of the equation for the eigenvalues; this diagram is a regular octagon. In various sectors of the indicator diagram we find the asymptotics for the eigenvalues of the studied differential operator.