Matrix Sturm-Liouville operators with boundary conditions dependent on the spectral parameter

Abstract

Let L denote the operator generated in L 2 (R + , E) by the differential expression l(y) = −y + Q(x)y, x ∈ R + , and the boundary condition (where Q is a matrix-valued function and A 0 , A 1 , B 0 , B 1 are non-singular matrices, with A 0 B 1 − A 1 B 0 = 0. In this paper, using the uniqueness theorems of analytic functions, we investigate the eigenvalues and the spectral singularities of L. In particular, we obtain the conditions on q under which the operator L has a finite number of the eigenvalues and the spectral singularities.

On Inverse Nodal Problem and Multiplicities of Eigenvalues of a Vectorial Sturm-Liouville Problem

“…Sturm-Liouville problems with eigenparameter in the boundary conditions arise upon separation of variables in the one-dimensional wave and heat equations for various physical applications [1]. There are some literatures on such scalar problems (see [2][3][4][5]) and vectorial problems (see [6][7][8]). We consider the inverse nodal problem of (1) firstly.…”

The Spectrum of Quadratic Eigenparameter-Dependent Non-Selfadjoint Matrix Sturm-Liouville Operators

“…All the above mentioned studies concern the spectral properties of the differential and difference equations with scalar coefficients. Nevertheless, the matter of the spectral theory of the differential and difference equations with matrix coefficients has also created interest among various authors [10][11][12][13][14]. In addition to that, inverse problem of matrix valued operators have been treated in [15][16][17][18].…”