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

Xiaoyun Liu

doi:10.1155/2020/6953602

Abstract

An m-dimensional vectorial inverse nodal Sturm-Liouville problem with eigenparameter-dependent boundary conditions is studied. We show that if there exists an infinite sequence ynj,rx,λnj,r2j=1∞ of eigenfunctions which are all vectorial functions of type (CZ), then the potential matrix Qx and A are simultaneously diagonalizable by the same unitary matrix U. Subsequently, some multiplicity results of eigenvalues are obtained.

Highlights

Consider the following m-dimensional vectorial SturmLiouville problem with the eigenparameter-dependent boundary conditions: 8 >>< −y′′ + QðxÞy = λ2y, ≤ x π,>>: yð0Þ = 0, Ay′ðπÞ +λyðπÞ ð1Þ where λ is the spectral parameter and y = ðy1, y2, ⋯, ymÞT is an m-dimensional vectorial function
We show that if there exists an infinite sequence fyn j,r ðx, λ2n j j=1 of eigenfunctions which are all vectorial functions of type (CZ), the potential matrix QðxÞ and A are simultaneously diagonalizable by the same unitary matrix U
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]

Summary

Introduction

Consider the following m-dimensional vectorial SturmLiouville problem with the eigenparameter-dependent boundary conditions:. 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]. In 1999, Shen and Shieh [18] studied the inverse nodal problem of the vectorial equation in (1) with Dirichlet boundary condition when dimension m = 2: They proved that if the problem has infinitely many eigenfunctions fyn j. It seems to be the first study of the vectorial inverse nodal problem. It is known that the eigenvalues of a scalar Sturm-Liouville problem with the separate boundary conditions are all simple. In [20], Shen and Shieh study the multiplicities of 2-dimensional vectorial Sturm-Liouville problem d2e×fi2neJdacionb[i0a,n1]m.

Preliminaries

Inverse Nodal Problem

Multiplicities of Eigenvalues