On a Spectral Property of Jacobi Matrices

Abstract

Let J be a Jacobi matrix with elements bk on the main diagonal and elements ak on the auxiliary ones. We suppose that J is a compact perturbation of the free Jacobi matrix. In this case the essential spectrum of J coincides with [−2, 2], and its discrete spectrum is a union of two sequences {x±j }, x+j ≥ 2, x−j ≤ −2, tending to ±2. We denote sequences {ak+1 − ak} and {ak+1 + ak−1 − 2ak} by ∂a and ∂a, respectively. The main result of the note is the following theorem. Theorem. Let J be a Jacobi matrix described above and σ be its spectral measure. Then a− 1, b ∈ l, ∂a, ∂b ∈ l, if and only if i) ∫ 2 −2 log σ(x)(4 − x) dx > −∞, ii) ∑

Cite this paper

@inproceedings{Kupin2007OnAS, title={On a Spectral Property of Jacobi Matrices}, author={S. Kupin}, year={2007} }