Spectral theory for a class of periodically perturbed unbounded Jacobi matrices: elementary methods

Authors:
Jan Janas;Serguei Naboko;Gunter Stolz
Affiliations:
Insitute of Mathematics, Polish Academy of Sciences, ul. sw. Tomasza 30, Krakow 31-027, Poland;Department of Mathematical Physics, Insitute of Physics St. Petersburg University, Ulianovkaia 1 198904 St. Petergoff St. Petersburg, Russia;Department of Mathematics, University of Alabama at Birmingham, CH 452, Birmingham, AL
Venue:
Journal of Computational and Applied Mathematics - Special issue: On the occasion of the eightieth birthday of prof. W.M. Everitt
Year:
2004

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Abstract

We use elementary methods to give a full characterization of the spectral properties of unbounded Jacobi matrices with zero diagonal and off-diagonal entries of the type λn = nα + cn, where ½ cn, is a real periodic sequence. The spectral properties depend strongly on the parity of the minimal period of Cn. The methods used are asymptotic diagonalization techniques, including the finite difference version of Levinson's theorem, subordinacy theory, and the variational principle.