Abstract
The eigenvalue problem of automorphic forms is the problem to determine the spectral behaviour of a certain second order elliptic partial differential operator having its domain of definition in an appropriate Hilbert space of square integrable functions defined in the upper halfplane. This Hilbert space depends on a discontinuous group G, a real parameter k, and a multiplier system v on G of weight 2k. We prove: The essential spectrum of the self-adjoint linear operator under consideration contains the interval\([\frac{1}{4}, \infty \rangle\) if G has a singular cusp or if G is a Fuchsian group of the second kind. If G has a sufficiently large fundamental domain, the essential spectrum of our differential operator is non-empty and unbounded above.
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Elstrodt, J., Roelcke, W. Über das wesentliche Spektrum zum Eigenwertproblem der automorphen Formen. Manuscripta Math 11, 391–406 (1974). https://doi.org/10.1007/BF01170240
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DOI: https://doi.org/10.1007/BF01170240