  In this paper we study the Laplace-Beltrami operator on quantum complex
hyperbolic spaces. We describe its action in terms of certain $q$-difference
operators of second order and prove spectral theorems for these operators. The
corresponding eigenfunctions are related to Al-Salam-Chihara polynomials. The
obtained Plancherel measure is related to a quantum analog for the
Harish-Chandra c-function.
