Quadratic eigenparameter dependent discrete Sturm–Liouville equations with spectral singularities

Abstract

Let us consider the boundary value problem (BVP) for the discrete Sturm–Liouville equation(0.1)an-1yn-1+bnyn+anyn+1=λyn,n∈N,(0.2)(γ0+γ1λ+γ2λ2)y1+(β0+β1λ+β2λ2)y0=0,where (an) and (bn),n∈N are complex sequences, γi,βi∈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, ifsupn∈Nexp(εnδ)1-an+bn<∞for some ε>0 and 12⩽δ⩽1.