The spectrum of eigenparameter-dependent discrete Sturm–Liouville equations

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