Asymptotics of the spectrum of a periodic boundary value problem for a differential operator with a summable potential

The spectrum of a differential operator of a high odd order with summable potential is studied. The boundary conditions are periodic. The differential equation that defines the differential operator is reduced to the Volterra integral equation. Solving this equation by Picard's method of successive approximations, we find the asymptotics of the fundamental system of solutions of the original differential equation. This fundamental system of solutions is used for the study of periodic boundary conditions. As a result, an equation for the eigenvalues of the differential operator is derived. This equation is a determinant of high order, which is an entire function of the spectral parameter. The indicator diagram corresponding to this function is investigated. The indicator diagram is a regular polygon and determines the location of the eigenvalues of the operator under consideration. As a result, the asymptotic behavior of the eigenvalues of the operator is found in each of the sectors of the complex plane determined by the indicator diagram (of 15th order).