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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



On selfadjoint extensions of some difference operator

Author: R. M. Kashaev
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 17 (2005), nomer 1.
Journal: St. Petersburg Math. J. 17 (2006), 157-167
MSC (2000): Primary 39A70, 47B25
Published electronically: 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 $ p$ and $ q$ 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 [Enhancements On Off] (What's this?)

  • 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)
  • 4. L. D. Faddeev and L. A. Takhtajan, Liouville model on the lattice, Field Theory, Quantum Gravity and Strings (Meudon/Paris, 1984/85), Lecture Notes in Phys., vol. 246, Springer, Berlin, 1986, pp. 166-179. MR 0848618 (87h:81213)
  • 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.
  • 7. R. M. Kashaev, Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998), 105-115; q-alg/9705021. MR 1607296 (99m:32021)
  • 8. -, The quantum dilogarithm and Dehn twists in quantum Teichmüller theory, Integrable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory (Kiev, 2000), NATO Sci. Ser. II Math. Phys. Chem., vol. 35, Kluwer Acad. Publ., Dordrecht, 2001, pp. 211-221. MR 1873573 (2003b:32017)
  • 9. J. Teschner, Liouville theory revisited, Classical Quantum Gravity 18 (2001), no. 23, R153-R222. MR 1867860 (2003f:81230)
  • 10. S. L. Woronowicz, Quantum exponential function, Rev. Math. Phys. 12 (2000), 873-920. MR 1770545 (2001g:47039)

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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

Keywords: Heisenberg commutation relation, discrete Liouville equation, selfadjoint extensions
Received by editor(s): September 15, 2004
Published electronically: 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
Article copyright: © Copyright 2006 American Mathematical Society

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