The Asymptotic Analysis of a Class of Self-Adjoint Second-Order Difference Equations: Jordan Box Case

Abstract

Abstract. In this paper, we are computing asymptotic formulas for a base of solutions of the second-order difference equations in the double root case. Two methods are presented.2000 AMS Subject Classifications. 39A10, 47B25.

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

“…And this could be the direct reason of gaps in proofs of non-subordinacy in many recent papers on spectral theory of Jacobi matrices: [4,8,17,19,26,27,33]. These papers concentrate mainly on the so-called Jordan box (i.e., critical) case-see [8].…”

“…We study various forms of asymptotics for two linearly independent generalised eigenvectors, found by the authors of [4,8,17,19,26,27,33] through some asymptotic tricks for difference equations. The results described below are not mutually independent-we start from some simpler cases to show the restrictions in use of some tools formulated in previous sections, and the reasons for formulating some others.…”

“…[8], because this is the main case when GBS method ([10, Lemma 1.5]) does usually not work. In particular, we try to repair the gaps in the proofs of non-subordinacy from [4,8,17,19,26,27,33].…”

“…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:…”

“…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]). …”