Spectrum of Discrete Second-Order Neumann Boundary Value Problems with Sign-Changing Weight

Ruyun Ma,Chenghua Gao,Yanqiong Lu

doi:10.1155/2013/280508

Abstract

We study the spectrum structure of discrete second-order Neumann boundary value problems (NBVPs) with sign-changing weight. We apply the properties of characteristic determinant of the NBVPs to show that the spectrum consists of real and simple eigenvalues; 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. We also show that the eigenfunction corresponding to thejth positive/negative eigenvalue changes its sign exactlyj-1times.

Highlights

Let n > 2 be an integer, T = {1, 2, . . . , n}
It is worth remarking that the number of sign changing of eigenfunction is given in Theorem 1
Since Jj is positive definite matrix for j = 1, 2, . . . , n−1 and J is positive semidefinite matrix, it follows that the roots of Qj(λ) are real, j = 1, 2, . . . , n

Summary

Introduction

It is worth remarking that the number of sign changing of eigenfunction is given in Theorem 1. N−1 and J is positive semidefinite matrix, it follows that the roots of Qj(λ) are real, j = 1, 2, .

Results

Conclusion