Density of the spectrum of Jacobi matrices with power asymptotics

Article type: Research Article

Authors: Pruckner, Raphael^{; *}

Affiliations: Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10/101, 1040 Wien, Austria. E-mail: raphael.pruckner@tuwien.ac.at

Correspondence: [*] Corresponding author. E-mail: raphael.pruckner@tuwien.ac.at.

Abstract: We consider Jacobi matrices J whose parameters have the power asymptotics ρn=nβ1(x0+x1n+O(n−1−ϵ)) and qn=nβ2(y0+y1n+O(n−1−ϵ)) for the off-diagonal and diagonal, respectively. We show that for β1>β2, or β1=β2 and 2x0>|y0|, the matrix J is in the limit circle case and the convergence exponent of its spectrum is 1/β1. Moreover, we obtain upper and lower bounds for the upper density of the spectrum. When the parameters of the matrix J have a power asymptotic with one more term, we characterise the occurrence of the limit circle case completely (including the exceptional case limn→∞|qn|/ρn=2) and determine the convergence exponent in almost all cases.

Keywords: Jacobi matrix, spectral analysis, difference equation, growth of entire function, canonical system, Berezanskiĭ’s theorem

DOI: 10.3233/ASY-191551

Journal: Asymptotic Analysis, vol. 117, no. 3-4, pp. 199-213, 2020