Spectral analysis of the matrix Sturm–Liouville operator

Abstract

The self-adjoint matrix Sturm-Liouville operator on a finite interval with a boundary condition in the general form is studied. We obtain asymptotic formulas for the eigenvalues and the weight matrices of the considered operator. These spectral characteristics play an important role in the inverse spectral theory. Our technique is based on analysis of analytic functions and on the contour integration in the complex plane of the spectral parameter. In addition, we adapt the obtained asymptotic formulas to the Sturm-Liouville operators on a star-shaped graph with two different types of matching conditions.

Spectral analysis of the Sturm‐Liouville operator on the star‐shaped graph

“…Proposition 2.1 has been proved in [45] for the general case and in [46] for the graph case. In order to provide asymptotic formulas for the weight matrices {α nk } n∈N, k=1,m , we need some additional notations.…”

“…Proposition 2.3 has been proved in [45] for the general case and in [39] for the graph case. Further we need the main equation of Inverse Problem 1.4, derived in [40].…”

“…The results of [39,40,45] yield the following necessary conditions on the spectral data. In the graph case, the solution of the main equation (2.12) is diagonal (Diagonality).…”

Constructive solution of the inverse spectral problem for the matrix Sturm–Liouville operator

“…In this section, we provide asymptotic formulas for the eigenvalues and the weight matrices, obtained in [2]. We also state the uniqueness theorem for solution of Inverse Problem 1.1.…”

“…In this section, we apply our results to the Sturm-Liouville eigenvalue problem on the starshaped graph [1,2] in the form − y j + q j (x)y j = λy j , x ∈ (0, π), j = 1, m, (5.1)…”