L2-Concrete Spectral Analysis of the Invariant LaplacianΔαβin the Unit Complex BallBn

Abstract

For every fixed real numberλrelated to the continuous spectrum of the invariant LaplaciansΔαβ=4(1−|z|2)∑i,j=1n(δij−zizj)∂2∂zj∂zj+α∑j=1nzj∂∂zj+β∑j=1nzj∂∂zj−αβinBn, we characterize the eigenfunctions ofΔαβthat are Poisson integrals ofL2-functions on the boundary ofBn. These eigenfunctions occurred in the weighted Plancherel formula of the unit complex ball. (See Zhang,Studia Math.102(2) (1992), for the weighted Plancherel formula in the ball.) The obtained characterization generalizes our announced result in Boussejra and Intissar,C. R. Acad. Sci.315(1992), 1353–1357, for the Bergman LaplacianΔ00of the unit complex ball to the LaplaciansΔαβ, for arbitraryα,βinR.