Spectral properties of Jacobi matrices by asymptotic analysis

Periodic perturbations of unbounded Jacobi matrices I: Asymptotics of generalized eigenvectors

“…Additive perturbations were investigated under the condition b n = 0. In [17], the periodic perturbations of n κ for κ ∈ (0, 1] were considered, namely, a n = n κ + d n , where (d n : n ∈ N) is a periodic sequence. Non-periodic perturbations were studied in [7].…”

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

“…3. In the proof of [8, Theorem 5.3., p. 617] the author refers to [11] for nonexistence of subordinate solutions. Unfortunately, this case cannot by covered by [11] (dealing only with "non-Jordan" 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]). …”