The spectrum of eigenparameter-dependent discrete Sturm-Liouville equations

摘要

Let us consider the boundary value problem (BVP) for the discrete SturmLiouville equation an-1yn-1+bny n+anyn+1=λ yn,n∈N, (γ0+γ1λ) y1+(β0+β1λ)y0=0, where (an) and (bn),n∈N are complex sequences, γi,βi∈C,i=0,1, and λ is a eigenparameter. Discussing the point spectrum, we prove that the BVP (0.1), (0.2) has a finite number of eigenvalues and spectral singularities with a finite multiplicities, if supn∈N[exp(ε nδ)(|1a n|+|bn|)]<∞, for some ε>0 and 12≤δ≤1.