\(L^2\)-concrete spectral analysis of the invariant Laplacian \(\Delta_{\alpha\beta}\) in the unit complex ball \(B^n\)
The operator \(\Delta_{\alpha\beta}\) in the title is \[ 4(1-| z| ^2) \biggl\{\sum^n_{i,j=1} (\delta_{ij}-z_i\overline z _j)\frac{\partial^2} {\partial z_j\partial\overline z _j} +\alpha\sum^n_{j=1} z_j \frac{\partial}{\partial z_j} +\beta\sum^n_{j=1}\overline z_j \frac{\partial} {\partial\overline z _j} -\alpha\beta\biggr\}, \] where \(\alpha\) and \(\beta\) are real numbers, and \(\delta_{ij}\) is the Kronecker symbol; and the ``concrete'' spectral theory is in the sense of \textit{R. S. Strichartz} [J. Funct. Anal. 87, No.1, 51-148 (1989; Zbl 0694.43008)]. The authors describe the eigenspaces of \(\Delta_{\alpha\beta}\) and characterize the eigenfunctions that are generalized Poisson transforms of square-integrable functions on the boundary of the unit ball in \(\mathbf{C}^n\). They previously announced the result in [C. R. Acad. Sci., Paris, Sér. I 315, No. 13, 1353-1357 (1992; Zbl 0772.32001)] for the special case when \(\alpha=\beta=0\).
scientific article; zbMATH DE number 1246379
scientific article