Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 063, 26 pages      arXiv:2102.13570      https://doi.org/10.3842/SIGMA.2021.063
Contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quantum in honor of Leon Takhtajan

Completeness of SoV Representation for $\mathrm{SL}(2,\mathbb R)$ Spin Chains

Sergey É. Derkachov a, Karol K. Kozlowski b and Alexander N. Manashov ca
a) St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia
b) Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France
c) Institut für Theoretische Physik, Universität Hamburg, D-22761 Hamburg, Germany

Received March 08, 2021, in final form June 14, 2021; Published online June 25, 2021

Abstract
This work develops a new method, based on the use of Gustafson's integrals and on the evaluation of singular integrals, allowing one to establish the unitarity of the separation of variables transform for infinite-dimensional representations of rank one quantum integrable models. We examine in detail the case of the $\mathrm{SL}(2,\mathbb R)$ spin chains.

Key words: spin chains; separation of variables; Gustafson's integrals.

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