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On selfadjoint extensions of some difference operator

Author(s): R. M. Kashaev
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 17 (2005), vypusk 1.
Journal: St. Petersburg Math. J. 17 (2006), 157-167.
MSC (2000): Primary 39A70, 47B25
Posted: January 19, 2006
MathSciNet review: 2140680
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Abstract | References | Similar articles | Additional information

Abstract: A one-parameter family of selfadjoint extensions is presented for the operator

$\displaystyle L=-e^{2\pi p}+2\cosh(z\pi bq),$

where $0<b\le 1$ and

and

are unbounded selfadjoint operators satisfying the Heisenberg commutation relation

$\displaystyle [p,q]=pq-qp=(2\pi i)^{-1}.$

The corresponding spectral problem is also solved.

References:

1.: L. O. Chekhov and V. V. Fock, Quantum Teichmüller spaces, Teoret. Mat. Fiz. 120 (1999), no. 3, 511-528; English transl., Theoret. and Math. Phys. 120 (1999), 1245-1259. MR 1737362 (2001g:32034)
2.: L. D. Faddeev, Quantum symmetry in conformal field theory by Hamiltonian methods, New Symmetry Principles in Quantum Field Theory (Cargèse, 1991) (J. Frölich et al., eds.), NATO Adv. Sci. Inst. Ser. B Phys., vol. 295, Plenum Press, New York, 1992, pp. 159-175. MR 1204454 (93k:81094)
3.: -, Discrete Heisenberg-Weyl group and modular group, Lett. Math. Phys. 34 (1995), 249-254. MR 1345554 (96i:46075)
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5.: L. D. Faddeev and A. Yu. Volkov, Algebraic quantization of integrable models in discrete space-time, Discrete Integrable Geometry and Physics (Vienna, 1996), Oxford Lecture Ser. Math. Appl., vol. 16, Oxford Univ. Press, New York, 1999, pp. 301-319; hep-th/97010039. MR 1676602 (2000h:81097)
6.: V. V. Fock, Dual Teichmüller spaces, Preprint dg-ga/9702018.
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Additional Information:

R. M. Kashaev
Affiliation: Université de Genève, Section de mathématiques, 2-4, rue du Lièvre, CP 240, 1211 Genève 24, Suisse
Email: Rinat.Kashaev@math.unige.ch

DOI: 10.1090/S1061-0022-06-00898-3
PII: S 1061-0022(06)00898-3
Keywords: Heisenberg commutation relation, discrete Liouville equation, selfadjoint extensions
Received by editor(s): 15/SEP/2004
Posted: January 19, 2006
Additional Notes: The author was supported in part by the Swiss National Science Foundation and by RFBR (grant no. 02-01-00085).
Dedicated: Dedicated to L. D. Faddeev on the occasion of his 70th birthday
Copyright of article: Copyright 2006, American Mathematical Society

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