Spectral properties of selfadjoint Jacobi matrices coming from birth and death processes

Abstract

Let J _a be the selfadjoint Jacobi matrix in l ²(ℕ) with the weights λ _n = n+a and the diagonal q _n = –2(n + a), a ∈ ℝ. If a ≥ 0 then J _a ≤ -I (the identity) and it is shown that its spectrum is purely absolutely continuous in (-∞, -1]. For a < 0 the point spectrum in (-1, +∞) is always nonempty. Moreover, J _a represents simple example of the spectral phase transition of the first order (see Introduction). The point spectrum of J _a (in particular its asymptotis behaviour as a tends to -∞) is also studied. The method of proof is based on finding asymptotic behaviour at {∞} of generalized eigenvectors of the equation J _a u=λ u, λ ∈ ℝ.