Spectrum of a Class of Difference Operators with Indefinite Weights

Congmin Yang,Gao Yun-Lan ,Kening Sun

doi:10.4236/jamp.2020.84056

Abstract

In this study, we use analytical methods and Sylvester inertia theorem to research a class of second order difference operators with indefinite weights and coupled boundary conditions. The eigenvalue problem with sign-changing weight has lasted a long time. The number of eigenvalues and the number of sign changes of the corresponding eigenfunctions of discrete equations under different boundary conditions are mainly studied. For the discrete Sturm-Liouville problems, similar conclusions about the properties of eigenvalues and the number of sign changes of the corresponding eigenfunctions are obtained under different boundary conditions, such as periodic boundary conditions, antiperiodic boundary conditions and separated boundary conditions etc. The purpose of this paper is to extend the similar conclusion to the coupled boundary conditions, which is of great significance to the perfection of the theory of the discrete Sturm-Liouville problems. We came to the following conclusions: first, the eigenvalues of the problem are real and single, the number of the positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. Second, under some conditions, we obtain the sign change of the eigenfunction corresponding to the j-th positive/negative eigenvalue.

Highlights

Spectral theory for Sturm-Liouville boundary value problems has important physical meaning and practical significance
We use analytical methods and Sylvester inertia theorem to research a class of second order difference operators with indefinite weights and coupled boundary conditions
The purpose of this paper is to extend the similar conclusion to the coupled boundary conditions, which is of great significance to the perfection of the theory of the discrete Sturm-Liouville problems

Summary

Introduction

Spectral theory for Sturm-Liouville boundary value problems has important physical meaning and practical significance. In 2007, Ji and Yang [4] [5] studied the structure of the eigenvalues of problem (1) (3) when the weight function m (t ) changes its sign, and they obtained that the number of positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. In 2013, Ma, Gao and Lu [7] discussed the spectra of the discrete second-order Neumann Eigenvalue problem (1) (6), By the analytical methods, he gives the properties of the eigenvalues, and gives the number of sign changes of the eigenfunction corresponding to the j-th positive/negative eigenvalue. We discuss different situations and get a series of important conclusions

Main Theorem

Lemma and Proof of the Theorem

Conclusion