Journal of Operator Theory

Volume 33, Issue 1, Winter 1995 pp. 159-196.

Spectral analysis of a Q-difference operator which arises from the quantum SU(1,1) group

Authors: Tomoyuki Kakehi (1), Tetsuya Masuda (2) and Kimio Ueno (3)
Author institution:(1) Institute of Mathematics, University of Tsukuba, Tsukuba 305, JAPAN
(2) Department of Mathematics, Waseda University, Tokyo 160, JAPAN

Summary: This paper is devoted to the study of an explicitly given second order difference operator which appears in the “representation theory” of the quantum SU (1, 1) group of non-compact type. We set up a situation in which the operator is shown to be self-adjoint, and the spectral analysis of the operator is developed. The “eigenfunctions” are perfectly given in terms of the basic hypergeometric functions. We then prove an explicit spectral expansion theorem which corresponds to the Fok-Mehler formula in the classical situation.

Keywords: Non-compact quantum groups, Zonal spherical functions, spectral theory of a self-adjoint operator.

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