Asymptotics of the spectrum of even-order differential operators with discontinuos weight functions

S. I. Mitrokhin

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 22 (2020) no. 1, pp. 48-70.

Résumé

The boundary-value problem for an eighth-order differential operator whose potential is a piecewise continuous function on the segment of the operator definition is studied. The weight function is piecewise constant. At the discontinuity points of the operator coefficients, the conditions of «conjugation» must be satislied which follow from physical considerations. The boundary conditions of the studied boundary value problem are separated and depend on several parameters. Thus, we simultaneously study the spectral properties of entire family of differential operators with discontinuous coefficients. The asymptotic behavior of the solutions of differential equations defining the operator is obtained for large values of the spectral parameter. Using these asymptotic expansions, the conditions of «conjugation» are investigated; as a result, the boundary conditions are studied. The equation on eigenvalues of the investigated boundary value problem is obtained. It is shown that the eigenvalues are the roots of some entire function. The indicator diagram of the eigenvalue equation is investigated. The asymptotic behavior of the eigenvalues in various sectors of the indicator diagram is found.

Détail

Keywords: boundary value problem, spectral parameter, differential operator, weight function, piecewise continuous potential, asymptotic behavior of eigenvalues.

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