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The central theme of this paper is the holomorphic spectral theory of thecanonical Laplace operator of the complement $\Omega := \{(z,w) \in\widehat{\mathbb{C}}^2 \colon z \cdot w \neq 1\}$ of the "complexified unitcircle" $\{(z,w) \in \widehat{\mathbb{C}}^2 \colon z \cdot w = 1\}$. We startby singling out a distinguished set of holomorphic eigenfunctions on the bidiskin terms of hypergeometric functions and prove that they provide a spectraldecomposition of every holomorphic eigenfunction on the bidisk. As a secondstep, we identify the maximal domains of definition of these eigenfunctions andshow that these maximal domains naturally determine the fine structure of theeigenspaces. Our main result gives an intrinsic classification of all closedM\"obius invariant subspaces of eigenspaces of the canonical Laplacian of$\Omega$. Generalizing foundational prior work of Helgason and Rudin, thisprovides a unifying complex analytic framework for the real-analytic eigenvaluetheories of both the hyperbolic and spherical Laplace operators on the openunit disk resp. the Riemann sphere and, in particular, shows how they areinterrelated with one another.

Spectral theory of the invariant Laplacian on the disk and the sphere -- a complex analysis approach