A class of $2n$th-order differential operator with eigenparameter-dependent boundary and transmission conditions

Abstract

In this paper, we investigate a class of 2nth-order regular differential operator with eigenparameter-dependent boundary conditions and transmission conditions at an interior discontinuous point. By constructing a new linear operator A associated with the problem, we prove that the operator A is self-adjoint in a suitable Hilbert space H , and the eigenvalues of the problem coincide with those of A. In terms of basic solutions of differential equation, we show that the eigenvalues of this problem coincide with the zeros of the entire function det˚.1; /, and obtain that the operator A has only point spectrum.