Non-Selfadjoin Difference Operators and Jacobi Matrices with Spectral Singularities

Abstract

In this paper we investigate the spectrum of the non-selfadjoint difference operatorL generated in l2(ℕ) by the difference expression (ly)n = an—1yn—1 + bnyn + anyn + 1 , n ∈ ℕ = {1 ,2, ...} and the boundary condition where a0 = 1, h0 ≠ 0 and are complex sequences and $\lbrace h_n \rbrace^\infty_{n = 1}$ ∈ l2(ℕ). We prove that L has the continuous spectrum, filling the segment [—2,2 ], a finite number of eigenvalues and spectral singularities with finite multiplicities if The results about the spectrum of L are applied to the non-selfadjoint Jacobi matrices and discreteSchrödinger operators.