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) byan-1yn-1 + bnyn + anyn+1 = λyn ; n ∈ ℕ; (0.1)pΣk=0 (y1γk + y0βk) λk = 0; (0.2)where {an} n∈ℕ and {bn} n∈ℕ are complex sequences, γi; βi ∈ ℂ; i = 0; 1; 2; ...; p andλ is a eigenparameter. Discussing the spectral properties of L, we investigate theJost function, spectrum, the spectral singularities, and the properties of the principalvectors corresponding to the spectral singularities of L, if∞Σn=1 n(∣1 - an∣ + ∣bnl) < ∞Mathematics Subject Classication (2010): 34L05, 34L40, 39A70, 47A10, 47A75.Key words: Discrete equations, eigenparameter, spectral analysis, eigenvalues, spectralsingularities, principal functions.