Cyclic Darboux 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), 795811
MSC (2000):
Primary 39A12, 39A13
Published electronically:
August 2, 2004
MathSciNet review:
2068795
Fulltext PDF Free Access
Abstract 
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Abstract: A discrete analog is constructed for the VeselovShabat 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 squareintegrable 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 .
 [1]
A.
B. Shabat and R.
I. Yamilov, Symmetries of nonlinear lattices, Algebra i Analiz
2 (1990), no. 2, 183–208 (Russian); 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, 96 (Russian, with
Russian summary); English transl., Funct. Anal. Appl. 27
(1993), no. 2, 81–96. MR 1251164
(94m:58179), http://dx.doi.org/10.1007/BF01085979
 [3]
V.
È. Adler, Nonlinear chains and Painlevé
equations, Phys. D 73 (1994), no. 4,
335–351. MR 1280883
(95c:58157), http://dx.doi.org/10.1016/01672789(94)90104X
 [4]
V.
M. Buchstaber and S.
P. Novikov (eds.), Solitons, geometry, and topology: on the
crossroad, American Mathematical Society Translations, Series 2,
vol. 179, American Mathematical Society, Providence, RI, 1997.
Advances in the Mathematical Sciences, 33. MR 1437154
(97i:00008)
 [5]
S.
P. Novikov and A.
P. Veselov, Exactly solvable twodimensional Schrödinger
operators and Laplace transformations, Solitons, geometry, and
topology: on the crossroad, 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 analogues of the harmonic oscillator,
Teoret. Mat. Fiz. 85 (1990), no. 1, 64–73
(Russian, with English summary); English transl., Theoret. and Math. Phys.
85 (1990), no. 1, 1055–1062 (1991). MR 1083952
(92i:81035), http://dx.doi.org/10.1007/BF01017247
 [7]
Vyacheslav
Spiridonov, Luc
Vinet, and Alexei
Zhedanov, Difference Schrödinger operators with linear and
exponential discrete spectra, Lett. Math. Phys. 29
(1993), no. 1, 63–73. MR 1242195
(94k:39031), http://dx.doi.org/10.1007/BF00760860
 [8]
Natig
M. Atakishiyev, Alejandro
Frank, and Kurt
Bernardo Wolf, A simple difference realization of the Heisenberg
𝑞algebra, J. Math. Phys. 35 (1994),
no. 7, 3253–3260. MR 1279301
(95c:81055), http://dx.doi.org/10.1063/1.530464
 [9]
I.
A. Dynnikov and S.
V. Smirnov, Exactly solvable cyclic Darboux 𝑞chains,
Uspekhi Mat. Nauk 57 (2002), no. 6(348),
183–184 (Russian); English transl., Russian Math. Surveys
57 (2002), no. 6, 1218–1219. MR 1991873
(2004f:37102), http://dx.doi.org/10.1070/RM2002v057n06ABEH000583
 [10]
S.
P. Novikov and I.
A. Dynnikov, Discrete spectral symmetries of smalldimensional
differential operators and difference operators on regular lattices and
twodimensional manifolds, Uspekhi Mat. Nauk 52
(1997), no. 5(317), 175–234 (Russian); English transl., Russian
Math. Surveys 52 (1997), no. 5, 1057–1116. MR 1490030
(99e:35029), http://dx.doi.org/10.1070/RM1997v052n05ABEH002105
 [1]
 A. B. Shabat and R. I. Yamilov, Symmetries of nonlinear lattices, Algebra i Analiz 2 (1990), no. 2, 183208; English transl., Leningrad Math. J. 2 (1991), no. 2, 377400. 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, 121; English transl., Funct. Anal. Appl. 27 (1993), no. 2, 8196. MR 1251164 (94m:58179)
 [3]
 V. E. Adler, Nonlinear chains and Painlevé equations, Phys. D 73 (1994), 335351. 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. 126130. MR 1437154 (97i:00008)
 [5]
 S. P. Novikov and A. P. Veselov, Exactly solvable twodimensional 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. 109132. 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, 6473; English transl., Theoret. and Math. Phys. 85 (1990), no. 1, 10551062 (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), 6773. 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), 32533260. MR 1279301 (95c:81055)
 [9]
 I. A. Dynnikov and S. V. Smirnov, Exactly solvable cyclic Darboux chains, Uspekhi Mat. Nauk 57 (2002), no. 6, 183184; 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 smalldimensional differential operators and difference operators on regular lattices and twodimensional manifolds, Uspekhi Mat. Nauk 52 (1997), no. 5, 175234; English transl., Russian Math. Surveys 52 (1997), no. 5, 10571116. 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:
http://dx.doi.org/10.1090/S1061002204008325
PII:
S 10610022(04)008325
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
