Automorphic Levi-Sobolev Spaces, Boundary-Value Problems, and Self-Adjoint Operators

Application of Plancherel's theorem to integral kernels approximating compact period functionals yields estimates on (global) automorphic Levi-Sobolev norms of the functionals. The utility of this viewpoint can be illustrated in reconsideration of several examples: Lax-Phillips' pseudo-Laplacians discretizing (part of) the continuous spectrum, Colin de Verdiere's meromorphic continuation of Eisenstein series, Hejhal's discussion of Haas' numerical analysis of the spectrum of the automorphic Laplacians, and construction of other self-adjoint operators on spaces of automorphic forms. Part of this work is joint with Enrico Bombieri.