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Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices

Janas, Jan, Naboko, Serguei, Stolz, Gunter (2009) Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices. International Mathematics Research Notices, 2009 (4). pp. 736-764. ISSN 1687-0247. (doi:10.1093/imrn/rnn144) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

 The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided. (Contact us about this Publication) Official URLhttp://dx.doi.org/10.1093/imrn/rnn144

Abstract

Bounds on the exponential decay of generalized eigenfunctions of bounded and unbounded selfadjoint Jacobi matrices in Graphic are established. Two cases are considered separately and lead to different results: (i) the case in which the spectral parameter lies in a general gap of the spectrum of the Jacobi matrix and (ii) the case of a lower semibounded Jacobi matrix with values of the spectral parameter below the spectrum. It is demonstrated by examples that both results are sharp. We apply these results to obtain a “many barriers-type” criterion for the existence of square-summable generalized eigenfunctions of an unbounded Jacobi matrix at almost every value of the spectral parameter in suitable open sets. In particular, this leads to examples of unbounded Jacobi matrices with a spectral mobility edge, i.e., a transition from purely absolutely continuous spectrum to dense pure point spectrum.

Item Type: Article 10.1093/imrn/rnn144 number of additional authors: 2; Q Science > QA Mathematics (inc Computing science) > QA276 Mathematical statistics Faculties > Sciences > School of Mathematics Statistics and Actuarial Science Stewart Brownrigg 07 Mar 2014 00:05 UTC 29 May 2019 12:25 UTC https://kar.kent.ac.uk/id/eprint/40458 (The current URI for this page, for reference purposes)
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