Relative oscillation theory for matrix Sturm-Liouville difference equations extended

Julia V Elyseeva

doi:10.1186/1687-1847-2013-328

Abstract

In this article we establish relative oscillation theorems for two discrete matrix Sturm-Liouville eigenvalue problems with Dirichlet boundary conditions and nonlinear dependence on the spectral parameter λ. This nonlinear dependence on λ is allowed both in the leading coefficients and in the potentials. Relative oscillation theory rather than measuring the spectrum of one single problem measures the difference between the spectra of two different problems. This is done by replacing focal points of conjoined bases of one problem by matrix analogs of weighted zeros of Wronskians of conjoined bases of two different problems.MSC: 39A21, 39A12.

Highlights

We consider the discrete matrix Sturm-Liouville spectral problemsRi(λ) xi(λ) – Qi(λ)xi+ (λ) =, i ∈ [, N – ], ( . )x (λ) = xN+ (λ) =, det Ri(λ) =, i ∈ [, N]and x (λ) = xN+ (λ) =, det Ri(λ) =, i ∈ [, N], where xi = xi+ – xi, xi(λ) ∈ Rn, n ≥, λ ∈ R is the spectral parameter, and the real symmetric n × n matrix-valued functions Ri(λ), Ri(λ), Qi(λ), Qi(λ), i ∈ [, N] are differentiable in the variable λ and obey the conditions d dλ Ri(λ) ≤, d dλ Qi(λ), i ∈ [, N], ( . ) d dλ
We investigate other representations of the relative oscillation numbers connected with different choices of symplectic difference systems associated with ( . ), ( . )
In Section we provide several examples illustrating the relative oscillation theory for scalar problems ( . ), ( . ) with nonlinear dependence on the spectral parameter

Summary

Introduction

In [ – ] we derive relative oscillation theory for symplectic difference eigenvalue problems with linear dependence on λ. We introduce the relative oscillation numbers which generalize the concept of a weighted zero of the Wronskian for the matrix case. According to [ , Section ], the number of focal points of a conjoined basis Yi = [XiT (Ri(λ) Xi)T ]T of matrix Sturm-Liouville equation

Results

Conclusion