Multipoint differential operators: ``splitting'' of the multiple in main eigenvalues

S. I. Mitrokhin

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 1, pp. 5-18

Résumé

We study the boundary value problem for the differential operator of the eighth order with a summable potential. The boundary conditions of the boundary value problem are multipoint. We derived the integral equation for solutions of differential equation which define the studied differential operator. The asymptotic formulas and estimates for the solutions of the corresponding differential equation for large values of the spectral parameter are obtained. By studying the boundary conditions, the equation for the eigenvalues as the determinant of the fourth order is derived. By using the properties of determinants and asymptotic formulas for solutions of differential equation we study the asymptotic behavior of the roots of the equation on eigenvalues of the operator. The coefficients of the boundary conditions of the studied boundary value problem are chosen in such a way that the main approach of the equation for the eigenvalues of the operator has two roots multiplicity three. The indicator diagram of the equation for the eigenvalues is studied in the detail. Studying one of the sectors of the indicator diagram, we derived the asymptotics of the eigenvalues of the studied operator. It is shown that the eigenvalues which are multiple in the main approximation “are split” into three single series of eigenvalues. Similar properties of eigenvalues are observed in other sectors of the indicator diagram.