Asymptotic behaviour of a differential operator with a finite number of transmission conditions

TL;DR

This paper studies the asymptotic behavior of eigenvalues and eigenfunctions for a class of discontinuous boundary value problems with transmission conditions, extending previous methods to finitely many discontinuities.

Contribution

It introduces a self-adjoint operator framework for these problems and derives asymptotic formulas, demonstrating the completeness of eigenfunctions in the Hilbert space.

Findings

Eigenvalues have specific asymptotic formulas.

Eigenfunctions form a complete set in the Hilbert space.

Extension of methods to finitely many discontinuities.

Abstract

In this paper following the same methods in [M. Kadakal, O. Sh. Mukhtarov, Sturm-Liouville problems with discontinuities at two points, Comput. Math. Appl., 54 (2007) 1367-1379] we investigate discontinuous two-point boundary value problems with eigenparameter in the boundary conditions and with transmission conditions at the finitely many points of discontinuity. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of such a problem coincide with those of A. We obtain asymptotic formulas for the eigenvalues and eigenfunctions. Also we show that the eigenfunctions of A are complete in H.