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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
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.
Received: 03.06.2014
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
Linking options:
https://www.mathnet.ru/eng/im8256https://doi.org/10.1070/IM2015v079n02ABEH002747 https://www.mathnet.ru/eng/im/v79/i2/p181
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Abstract page: | 1166 | Russian version PDF: | 290 | English version PDF: | 28 | References: | 122 | First page: | 95 |
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