Spectra of some selfadjoint Jacobi operators in the double root case

Abstract

Abstract. In this paper we prove a mixed spectrum of Jacobi operators defined by λn = s(n)(1 + x(n)) and qn = −2s(n)(1 + y(n)), where (s(n)) is a real unbounded sequence, (x(n)) and (y(n)) are some perturbations.

Orthogonal Polynomials with Periodically Modulated Recurrence Coefficients in the Jordan Block Case II

“…[5-7, 10, 11, 17, 18, 23, 43, 54, 55, 60]; (b) if X 0 (0) is not diagonalizable then the only situation which was known is the case when either the essential spectrum of A is empty or it is a half-line, see e.g. [4,8,9,20,22,33,34,36,37,39,44,51,61,66]; III. if | tr X 0 (0)| > 2, then under some regularity assumptions on Jacobi parameters the essential spectrum of A is empty, see e.g.…”

About essential spectra of unbounded Jacobi matrices

“…It is also rich enough to allow building an intuition about the general case. In particular, in this class there are examples of Jacobi matrices with purely absolutely continuous spectrum filling the whole real line (see [15,17,35,38,41]), having a bounded gap in absolutely continuous spectrum (see [5-7, 9, 10, 12, 14, 19, 27]), having absolutely continuous spectrum on the half-line (see [4,8,16,18,[23][24][25][26]33]), having purely singular continuous spectral measure with explicit Hausdorff dimension (see [2]), having a dense point spectrum on the real line (see [2]), and having an empty essential spectrum (see [11,[29][30][31]42]).…”

Periodic perturbations of unbounded Jacobi matrices III: The soft edge regime

“…We have to show that B restricted to ℓ 2 (N; C N ) has the requested properties. Decompose operator B by(21)…”

“…In [5,20,21,22,28] were considered some special cases corresponding to the more challenging setting of the hard edge regime (see Definition 1).…”