Spectral Analysis of Non-Selfadjoint Discrete Schrödinger Operators with Spectral Singularities

Let L denote the non-selfadjoint discrete Schrödinger operator generated in 𝓁₂(ℕ) by the difference expression

(𝓁y)_n = y_{n –1} + y_{n + 1} + b_ny_n, n ∈ ℕ = {1, 2, …,}

and the boundary condition y₀ = 0, where {b_n}^∞_{n = 1} is a complex sequence. In this paper we investigate Weyl-Titchmarsh (W – T ) function of the operator L and obtained the relation between W – T function and the generalized spectral function of L in the sense of Marchenko. Moreover we find Cauchy type integral representation of W – T function. Using this representation we derived the spectral expansion of L in terms of the principal vectors, taking into account the spectral singularities.