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Izvestiya: Mathematics, 2015, Volume 79, Issue 2, Pages 388–410
DOI: https://doi.org/10.1070/IM2015v079n02ABEH002747
(Mi im8256)
 

This article is cited in 17 scientific papers (total in 17 papers)

The spectral theory of a functional-difference operator in conformal field theory

L. A. Takhtadzhyanab, L. D. Faddeevcd

a Euler International Mathematical Institute, St. Petersburg
b Department of Mathematics, Stony Brook University
c Saint Petersburg State University
d St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences
References:
Abstract: We consider the functional-difference operator $H=U+U^{-1}+V$, where $U$ and $V$ are the Weyl self-adjoint operators satisfying the relation $UV=q^{2}VU$, $q=e^{\pi i\tau}$, $\tau>0$. The operator $H$ has applications in the conformal field theory and representation theory of quantum groups. Using the modular quantum dilogarithm (a $q$-deformation of the Euler dilogarithm), we define the scattering solution and Jost solutions, derive an explicit formula for the resolvent of the self-adjoint operator $H$ on the Hilbert space $L^{2}(\mathbb R)$, and prove the eigenfunction expansion theorem. This theorem is a $q$-deformation of the well-known Kontorovich–Lebedev transform in the theory of special functions. We also present a formulation of the scattering theory for $H$.
Keywords: modular quantum dilogarithm, Weyl operators,functional-difference operator, Schrödinger operator, Fourier transform, Casorati determinant, Sokhotski–Plemelj formula, scattering solution, Jost solutions, resolvent of an operator, eigenfunction expansion, Kontorovich–Lebedev transform, scattering theory, scattering operator.
Funding agency Grant number
National Science Foundation DMS-1005769
Russian Foundation for Basic Research 14-01-00341
13-01-12405-офи-м
Russian Academy of Sciences - Federal Agency for Scientific Organizations
The first author's work was partially supported by NSF grant no. DMS-1005769. The second author's work was partially supported by RFBR grants nos. 14-01-00341, 13-01-12405-ofi-m and by the RAS programme "Mathematical Problems of Non-linear Dynamics".
Received: 03.06.2014
Bibliographic databases:
Document Type: Article
UDC: 517.581+517.965+517.984
MSC: 33D05, 34K06, 39A70
Language: English
Original paper language: Russian
Citation: L. A. Takhtadzhyan, L. D. Faddeev, “The spectral theory of a functional-difference operator in conformal field theory”, Izv. Math., 79:2 (2015), 388–410
Citation in format AMSBIB
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\by L.~A.~Takhtadzhyan, L.~D.~Faddeev
\paper The spectral theory of a~functional-difference operator in conformal field theory
\jour Izv. Math.
\yr 2015
\vol 79
\issue 2
\pages 388--410
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  • https://doi.org/10.1070/IM2015v079n02ABEH002747
  • https://www.mathnet.ru/eng/im/v79/i2/p181
  • This publication is cited in the following 17 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Abstract page:1166
    Russian version PDF:290
    English version PDF:28
    References:122
    First page:95
     
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