S.I. Mitrokhin. Asymptotics of the spectrum of a periodic boundary value problem for a differential operator with a summable potential ... P. 136-149

Vol. 25, no. 1, 2019

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).

Keywords: spectral parameter, differential operator, summable potential, periodic boundary conditions, asymptotics of solutions of a differential equation, asymptotics of the spectrum

Received September 18, 2018

Revised November 5, 2018

Accepted November 12, 2018

Sergey Ivanovich Mitrokhin, Cand. Sci. (Phys.-Math.), Prof. of the Russian Academy of Natural Sciences, Scientific-Research Computing Center of the Moscow State University (Lomonosov), Moscow, 119991 Russia, e-mail: mitrokhin-sergey@yandex.ru

Cite this article as: S.I. Mitrokhin. Asymptotics of the spectrum of a periodic boundary value problem for a differential operator with a summable potential, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 1, pp.  136-149. 

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