Quadratic eigenparameter dependent discrete Sturm-Liouville equations with spectral singularities

Abstract

Let us consider the boundary value problem (BVP) for the discrete Sturm-Liouville equationan-1yn-1+bnyn+anyn+1=λyn,n N,( γ0+γ1λ+γ2λ2)y1+(β0+ β1λ+β2λ2) y0=0,where (an) and (bn),nâ̂̂N are complex sequences, γi,βiâ̂̂ C,i=0,1,2, and λ is a eigenparameter. Discussing the point spectrum, we prove that the BVP (0.1) and (0.2) has a finite number of eigenvalues and spectral singularities with a finite multiplicities, ifsupnâ̂̂ Nexp(εnδ)1-an+bn<â̂ for some ε>0 and 12≤δ≤1. © 2014 Elsevier Inc. All rights reserved.