Spectral properties of generalized eigenparameter dependent discrete Sturm-Liouville type equation

Abstract

Let L denote the non-selfadjoint difference operator of second order with boundary condition generated in ℓ₂ (ℕ) by

a_n-1y_n-1 + b_ny_n + a_ny_n+1 = λy_n ; n ∈ ℕ;   (0.1)
^pΣ_k=0 (y1γ_k + y0β_k) λ^k = 0;                    (0.2)

where {a_n} _n∈ℕ and {b_n} _n∈ℕ are complex sequences, γi; βi ∈ ℂ; i = 0; 1; 2; ...; p and
λ is a eigenparameter. Discussing the spectral properties of L, we investigate the
Jost function, spectrum, the spectral singularities, and the properties of the principal
vectors corresponding to the spectral singularities of L, if

^∞Σ_n=1  n(∣1 - a_n∣ + ∣b_nl) < ∞

Mathematics Subject Classication (2010): 34L05, 34L40, 39A70, 47A10, 47A75.

Key words: Discrete equations, eigenparameter, spectral analysis, eigenvalues, spectral
singularities, principal functions.