The spectrum of eigenparameter-dependent discrete Sturm–Liouville equations

Let us consider the boundary value problem (BVP) for the discrete Sturm–Liouville equation (0.1) a n − 1 y n − 1 + b n y n + a n y n + 1 = λ y n , n ∈ N , (0.2) ( γ 0 + γ 1 λ ) y 1 + ( β 0 + β 1 λ ) y 0 = 0 , where ( a n ) and ( b n ) , n ∈ N are complex sequences, γ i , β i ∈ 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 sup n ∈ N [ exp ( ε n δ ) ( | 1 − a n | + | b n | ) ] < ∞ , for some ε > 0 and 1 2 ≤ δ ≤ 1 .