Inequalities among eigenvalues of different self-adjoint discrete Sturm-Liouville problems

TL;DR

This paper establishes inequalities among eigenvalues of self-adjoint discrete Sturm-Liouville problems, considering variations in boundary conditions and equations, generalizing previous results in the field.

Contribution

It introduces new inequalities relating eigenvalues for different boundary conditions and equations, expanding the understanding of eigenvalue behavior in discrete Sturm-Liouville problems.

Findings

Inequalities among eigenvalues for different boundary conditions.

Inequalities among eigenvalues for different equations.

Generalization of existing eigenvalue inequality results.

Abstract

In this paper, inequalities among eigenvalues of different self-adjoint discrete Sturm-Liouville problems are established. For a fixed discrete Sturm-Liouville equation, inequalities among eigenvalues for different boundary conditions are given. For a fixed boundary condition, inequalities among eigenvalues for different equations are given. These results are obtained by applying continuity and discontinuity of the n-th eigenvalue function, monotonicity in some direction of the n-th eigenvalue function, which were given in our previous papers, and natural loops in the space of boundary conditions. Some results generalize the relevant existing results about inequalities among eigenvalues of different Sturm-Liouville problems.