Spectral theory of certain unbounded Jacobi matrices

Abstract

Using the conjugate operator method of Mourre we study the spectral theory of a class of unbounded Jacobi matrices. We especially focus on the case where the off-diagonal entries a n = n α (1 + o(1)) and diagonal ones b n = λn α (1 + o(1)) with α > 0, λ ∈ R.

Boundary pairs associated with quadratic forms

“…This ansatz allows us to use known results on the spectrum of this special Jacobi operator (see e.g. [, Thm 1.1] and references therein; as well as for the case and the general ideas of the spectral analysis). The spectrum of J is purely discrete if , and absolutely continuous if .…”

Spectral Properties of Unbounded Jacobi Matrices with Almost Monotonic Weights

“…A method often used is based on subordination theory (see, e.g., [6,15,19]). Another technique uses the analysis of a commutator between a Jacobi matrix and a suitable chosen matrix (see, e.g., [22]). The case of Jacobi matrices with monotonic weights was considered mainly by Dombrowski (see, e.g., [8]), where the author developed commutator techniques which enabled qualitative spectral analysis of examined operators.…”

On the spectrum of periodic perturbations of certain unbounded Jacobi operators

“…(ii) The assertion (b1) extends Theorem 1.5 of page 507 of [1] which only covers the case where λ = 0, for more details see [13]. (iii) The assertion (b2) can be deduced from [6,7], see also [13].…”

“…(iii) The assertion (b2) can be deduced from [6,7], see also [13]. (iv) The point (c1) follows from Theorem 8 of [5].…”