Cyclic Darboux 𝑞-chains

Smirnov, S.

doi:10.1090/S1061-0022-04-00832-5

Cyclic Darboux $q$-chains
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by S. V. Smirnov
Translated by: B. M. Bekker

St. Petersburg Math. J. 15 (2004), 795-811

DOI: https://doi.org/10.1090/S1061-0022-04-00832-5

Published electronically: August 2, 2004

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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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