Spectral Analysis of Selfadjoint Jacobi Matrices with Periodically Modulated Entries

Abstract

This paper deals with the spectral analysis of a class of selfadjoint unbounded Jacobi matrices J with modulated entries. The entries have the form of smooth sequences that increase to infinity multiplied by proper periodic sequences. For this class criteria for pure absolute continuity of the spectrum or its discreteness, and the asymptotics of generalized eigenvectors of J, are given. Some examples illustrating the stability zones of spectral structure are presented.

Periodic perturbations of unbounded Jacobi matrices II: Formulas for density

“…and (b), by (14) and (16) one gets discr F K (x) → discr F almost uniformly. Therefore, the right-hand side of the numerator of (13) converges to √ − discr F. Consequently, (3) implies that there exists K 0 such that for all K ≥ K 0 one has I ⊂ E K .…”

“…[3,5,6,12,13,14,25]. As far as the approximation of µ is concerned, the only result known to the author is [1].…”

Non-Existence of Subordinate Solutions for Jacobi Operators in Some Critical Cases

“…Subordination theory, formulated first by Gilbert and Pearson in [7] for one dimensional Shrödinger operators, where transferred to the discrete case by [22,34], became a very popular method used for spectral analysis of Jacobi operators in 2 (N) (see, e.g., [3][4][5][8][9][10][11][12][13][14][15][16][17][18][19][20][21][23][24][25][26][27]29,30,32,33]). One of the typical ways of employing the subordination theory for spectral studies is related to asymptotic studies, and it contains the following three basic steps:…”

“…Finding asymptotics of some solutions to the generalised eigenequation (generalised eigenvectors) for the Jacobi operator J. This is often achieved with the use of some discrete Levinson type theorems-see [3,5,9,[11][12][13]16,18,19,21,24,25,27,30,32]; 1 2. Reading some information on the existence of subordinate solution (see Definition (2.1)) from the asymptotic information on solutions; 3.…”

“…Such schema was particularly fruitful for studies of the absolute continuous spectrum, where the absence of the subordinate solution plays a crucial role (e.g., [3][4][5]8,9,[11][12][13][14][15][17][18][19][24][25][26][27]33]). …”