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

TL;DR

This paper develops a complex analysis framework to study the spectral theory of the invariant Laplacian on the disk and sphere, revealing new classifications of eigenspaces and unifying real-analytic eigenvalue theories.

Contribution

It introduces a holomorphic spectral theory for the invariant Laplacian on a complex domain, classifies M"obius invariant eigenspaces, and unifies hyperbolic and spherical Laplacian eigenvalue theories.

Findings

Holomorphic eigenfunctions are characterized by hypergeometric functions.

Maximal domains of eigenfunctions determine eigenspace structures.

Provides a unified complex analytic framework linking hyperbolic and spherical Laplacians.

Abstract

The central theme of this paper is the holomorphic spectral theory of the canonical Laplace operator of the complement $\Omega := \{(z,w) \in \widehat{\mathbb{C}}^2 \colon z \cdot w \neq 1\}$ of the "complexified unit circle" $\{(z,w) \in \widehat{\mathbb{C}}^2 \colon z \cdot w = 1\}$. We start by singling out a distinguished set of holomorphic eigenfunctions on the bidisk in terms of hypergeometric functions and prove that they provide a spectral decomposition of every holomorphic eigenfunction on the bidisk. As a second step, we identify the maximal domains of definition of these eigenfunctions and show that these maximal domains naturally determine the fine structure of the eigenspaces. Our main result gives an intrinsic classification of all closed M\"obius invariant subspaces of eigenspaces of the canonical Laplacian of $\Omega$. Generalizing foundational prior work of Helgason and Rudin, this provides a unifying complex analytic framework for the real-analytic eigenvalue theories of both the hyperbolic and spherical Laplace operators on the open unit disk resp. the Riemann sphere and, in particular, shows how they are interrelated with one another.