# 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 â€œrepresentation theoryâ€ 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 â€œeigenfunctionsâ€ 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.

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