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Cyclic Darboux  -chains
Author(s):
S.
V.
Smirnov
Translated by:
B. M. Bekker
Original publication:
Algebra i Analiz,
tom 15
(2003),
vypusk 5.
Journal:
St. Petersburg Math. J.
15
(2004),
795-811.
MSC (2000):
Primary 39A12, 39A13
Posted:
August 2, 2004
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Additional information
Abstract:
A discrete  -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  -arithmetic progressions. Any cyclic  -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  -oscillator constructed in the paper weakly converges to the usual harmonic oscillator as  .
References:
-
- [1]
- A. B. Shabat and R. I. Yamilov, Symmetries of nonlinear lattices, Algebra i Analiz 2 (1990), no. 2, 183-208; English transl., Leningrad Math. J. 2 (1991), no. 2, 377-400. MR 1062269 (91k:58116)
- [2]
- A. P. Veselov and A. B. Shabat, A dressing chain and the spectral theory of the Schrödinger operator, Funktsional. Anal. i Prilozhen. 27 (1993), no. 2, 1-21; English transl., Funct. Anal. Appl. 27 (1993), no. 2, 81-96. MR 1251164 (94m:58179)
- [3]
- V. E. Adler, Nonlinear chains and Painlevé equations, Phys. D 73 (1994), 335-351. MR 1280883 (95c:58157)
- [4]
- S. P. Novikov and I. A. Taimanov, Difference analogs of the harmonic oscillator, Appendix II in [5], Solitons, Geometry, and Topology: on the Crossroad, Amer. Math. Soc. Transl. Ser. 2, vol. 179, Amer. Math. Soc., Providence, RI, 1997, pp. 126-130. MR 1437154 (97i:00008)
- [5]
- S. P. Novikov and A. P. Veselov, Exactly solvable two-dimensional Schrödinger operators and Laplace transformations, Solitons, Geometry, and Topology: on the Crossroad (V. M. Buchstaber, S. P. Novikov, eds.), Amer. Math. Soc. Transl. Ser. 2, vol. 179, Amer. Math. Soc., Providence, RI, 1997, pp. 109-132. MR 1437160 (98a:58088)
- [6]
- N. M. Atakishiev and S. K. Suslov, Difference analogs of the harmonic oscillator, Teoret. Mat. Fiz. 85 (1990), no. 1, 64-73; English transl., Theoret. and Math. Phys. 85 (1990), no. 1, 1055-1062 (1991). MR 1083952 (92i:81035)
- [7]
- V. Spiridonov, L. Vinet, and A. Zhedanov, Difference Schrödinger operators with linear and exponential discrete spectra, Lett. Math. Phys. 29 (1993), 67-73. MR 1242195 (94k:39031)
- [8]
- N. Atakishiev, A. Frank, and K. Wolf, A simple difference realization of the Heisenberg
 -algebra, J. Math. Phys. 35 (1994), 3253-3260. MR 1279301 (95c:81055) - [9]
- I. A. Dynnikov and S. V. Smirnov, Exactly solvable cyclic Darboux
 -chains, Uspekhi Mat. Nauk 57 (2002), no. 6, 183-184; English transl. in Russian Math. Surveys 57 (2002), no. 6. MR 1991873 (2004f:37102) - [10]
- S. P. Novikov and I. A. Dynnikov, Discrete spectral symmetries of small-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds, Uspekhi Mat. Nauk 52 (1997), no. 5, 175-234; English transl., Russian Math. Surveys 52 (1997), no. 5, 1057-1116. MR 1490030 (99e:35029)
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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:
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):
31/JUL/2002
Posted:
August 2, 2004
Copyright of article:
Copyright
2004,
American Mathematical Society
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