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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Cyclic Darboux $q$-chains


Author: S. V. Smirnov
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 15 (2003), nomer 5.
Journal: St. Petersburg Math. J. 15 (2004), 795-811
MSC (2000): Primary 39A12, 39A13
Published electronically: August 2, 2004
MathSciNet review: 2068795
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Abstract: A discrete $q$-analog is constructed for the Veselov-Shabat dressing chain (the latter is a generalization of the classical harmonic oscillator). It is shown that, as in the continuous case, the corresponding operator relations make it possible to completely determine the discrete spectra of the operators in the chain: each spectrum consists of several $q$-arithmetic progressions. Any cyclic $q$-chain can be realized by bounded selfadjoint difference operators, the spectrum of each of them is discrete, and the eigenvectors form a complete family in the Hilbert space of square-integrable sequences. Moreover, an explicit general solution is given for chains of length 2, and it is proved that the $q$-oscillator constructed in the paper weakly converges to the usual harmonic oscillator as $q\to 1$.


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Additional Information

S. V. Smirnov
Affiliation: Moscow State University, Department of Mathematics and Mechanics, Moscow, 119992, Russia
Email: sergey@svsmir.mccme.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-04-00832-5
PII: S 1061-0022(04)00832-5
Keywords: Harmonic oscillator, $q$-oscillator, Darboux $q$-chain
Received by editor(s): July 31, 2002
Published electronically: August 2, 2004
Article copyright: © Copyright 2004 American Mathematical Society