Asymptotics of the spectrum of a periodic boundary value problem for an odd-order differential operator

The spectrum of a differential operator of high odd order with periodic boundary conditions is studied. The asymptotics of the fundamental system of solutions of the differential equation defining the operator are obtained by the method of successive Picard approximations. With the help of this fundamental system of solutions the periodic boundary conditions are studied. As a result, the equation for the eigenvalues of the differential operator is obtained, which is quasi-polynomial. The indicator diagram of this equation, which is a regular polygon, is investigated. In each of the sectors of the complex plane, defined by the indicator diagram, the asymptotics of the eigenvalues of the operator is found. An equation for the eigenvalues of the differential operator is derived. The indicator diagram of this equation has been studied. The asymptotics of the eigenvalues of the studied operator in different sectors of the indicator diagram is found.