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On the Spectral Properties of a Multipoint Boundary Value Problem for an Odd-Order Differential Operator with Summable Potential. P. 376–392

Версия для печати

Section: Physics. Mathematics. Informatics

UDC

517.927

Authors

Sergey I. Mitrokhin*
*Lomonosov Moscow State University (Moscow, Russian Federation)
Corresponding author: Sergey Mitrokhin, address: MPO-1, Leninskie gory, 1, stroenie 4, Moscow, 119991, Russian Federation; e-mail: mitrokhin-sergey@yandex.ru

Abstract

The paper studies the boundary value problem for a high odd order differential operator. The potential of an operator is a summable function on the segment of the operator’s study. The boundary conditions are given on the boundaries of the segment and at several interior points that divide the segment into incommensurable parts. Thus, the boundary conditions are multipoint. Multipoint boundary conditions arise when studying the vibrations of bridges and beams, the bearings of which are located at internal points. The paper demonstrates the asymptotics of solutions of the corresponding differential equation for large values of the spectral parameter under the condition of potential summability. Previously, the asymptotics of solutions of differential equations was studied in the case of smooth coefficients and piecewise smooth coefficients. Asymptotic estimates in various sectors of the complex plane are obtained similarly to the derivation of estimates by the method of M.A. Naimark. Using the obtained asymptotics of the solutions, boundary conditions are investigated. This study leads to a system of homogeneous equations that has non-trivial solutions only when its determinant is zero. Thus, an equation is obtained which is satisfied by the eigenvalues of the studied operator. We investigate the indicator diagram of this equation. The function that the eigenvalues satisfy is an entire in various sectors of the indicator diagram. Using the indicator diagram, we find the asymptotics of the eigenvalues of the studied differential operator. The spectrum of the operator is discrete. This operator does not exhibit the effect of a “splitting” of multiples in the principal eigenvalues. We can investigate the behavior of eigenfunctions of the studied operator using the obtained spectrum.

Keywords

multipoint boundary value problem, spectral parameter, multipoint boundary conditions, summable potential, indicator diagram, asymptotics of eigenvalues
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