About essential spectra of unbounded Jacobi matrices

Abstract

GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJAN A. We study spectral properties of unbounded Jacobi matrices with periodically modulated or blended entries. Our approach is based on uniform asymptotic analysis of generalized eigenvectors. We determine when the studied operators are self-adjoint. We identify regions where the point spectrum has no accumulation points. This allows us to completely describe the essential spectrum of these operators.

Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case II

“…To prove Theorem A it is sufficient to analyze the asymptotic behavior of generalized eigenvectors. In Theorem 4.1, with a help of a recently obtained variant of discrete Levinson's type theorem (see [23]), we show that (1.5) holds for every compact interval ⊂ Λ + . The fact that (1.3) holds for every compact interval ⊂ Λ − is a consequence of the following theorem.…”

“…Since [23,Theorem 4.4] allows perturbation satisfying (10.3) we can repeat the proof of Theorem 4.1 to the get following statement.…”

“…if | tr 0 (0)| = 2, then we have two subcases: a) if 0 (0) is diagonalizable, then, under some regularity assumptions on Jacobi parameters, there is a compact interval ⊂ R such that is purely absolutely continuous on R \ and it is purely discrete in the interior of , see e.g. [23]; b) if 0 (0) is not diagonalizable, then the only situation which was known is the case when either the essential spectrum of is empty or it is a half-line, see e.g. [25,28]; III.…”

“…if | tr 0 (0)| > 2, then, under some regularity assumptions on Jacobi parameters, the essential spectrum of is empty, see e.g. [10,23].…”

“…Observe that, if = 0 then at least one of the sets Λ − or Λ + is empty. Similar to the proof of[25, Theorem 4.1] we use the shifted conjugation (Theorems 3.1 and 3.2) together with a variant of Levin's theorem developed in[23, Theorem 4.4].Theorem 4.1. Let be a positive integer.…”