Spectrum of a Class of Difference Operators with Indefinite Weights

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.

Reproducing Kernel Function for Sturm-Liouville Problem with Variable Coefficients

“…As of late, periodic boundary value problem of two-point second-order mixed integro-differential equation is tackled in [14][15][16][17][18]. A class of second order difference operators with indefinite weights and coupled boundary conditions is treated in [19][20]. The present paper is structured as follows: Section 2 is dedicated to introduce the principle properties of the class of the Hilbert space theory and suggested RKHS.…”