Harmonic Analysis on Quantum Complex Hyperbolic Spaces
Olga Bershtein, Yevgen Kolisnyk

TL;DR
This paper develops harmonic analysis on quantum complex hyperbolic spaces by introducing a quantum Laplace-Beltrami operator, analyzing its spectral properties, and establishing a Plancherel theorem, thus extending classical harmonic analysis to a quantum setting.
Contribution
It introduces a quantum analog of the Laplace-Beltrami operator and analyzes its spectral properties, including eigenfunctions related to Al-Salam-Chihara polynomials.
Findings
Eigenfunctions related to Al-Salam-Chihara polynomials
Second order q-difference operator identified
Plancherel theorem established for the quantum setting
Abstract
In this paper we obtain some results of harmonic analysis on quantum complex hyperbolic spaces. We introduce a quantum analog for the Laplace-Beltrami operator and its radial part. The latter appear to be second order -difference operator, whose eigenfunctions are related to the Al-Salam-Chihara polynomials. We prove a Plancherel type theorem for it.
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