Spectral analysis of the matrix Sturm\u2013Liouville operator

Natalia P Bondarenko

doi:10.1186/s13661-019-1292-z

Natalia P Bondarenko

Open Access

https://doi.org/10.1186/s13661-019-1292-z

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Abstract

The self-adjoint matrix Sturm–Liouville operator on a finite interval with a boundary condition in 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 an 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.

Highlights

1 Introduction The paper concerns the spectral theory of differential operators
We focus on analysis of matrix Sturm–Liouville operators in the form Y = –Y + Q(x)Y, where Q(x) is a matrix function
The latter operators are often called quantum graphs. They are applied for modeling wave propagation through a domain being a thin neighborhood of a graph. Such models are used in organic chemistry, mesoscopic physics, nanotechnology and other branches of science and engineering

Summary

Introduction

The paper concerns the spectral theory of differential operators. In particular, we focus on analysis of matrix Sturm–Liouville operators in the form Y = –Y + Q(x)Y , where Q(x) is a matrix function. In [15,16,17,18], inverse spectral problems have been studied, which consist in a reconstruction of the matrix Sturm–Liouville operators by using the eigenvalues and the weight matrices. Xu [19] proved uniqueness theorems for inverse problems for Eq (1.1) with the two boundary conditions in the general self-adjoint form Such operators are especially worth to be studied because of their applications to quantum graphs. Lemma 2.4 The problem L has a countable set of eigenvalues {λnk}n∈N,k=1,m, λnk = ρn2k, numbered according to their multiplicities and having the following asymptotics:. Applying Proposition 2.2, we conclude that, for sufficiently large n, the function det(R(z)) has exactly (m – p) zeros {znk}k=p+1,m (counted according to their multiplicities), having the asymptotics znk = zk + nk, nk = o(1), n → ∞, k = p + 1, m.

The residue theorem yields

Proof Consider the set of indices

Then ρnk ηnk n

The matrix