Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Scattering theory for
twisted automorphic functions

Author: Ralph Phillips
Journal: Trans. Amer. Math. Soc. 350 (1998), 2753-2778
MSC (1991): Primary 58G25, 11F72, 35L05
MathSciNet review: 1466954
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Abstract: The purpose of this paper is to develop a scattering theory for twisted automorphic functions on the hyperbolic plane, defined by a cofinite (but not cocompact) discrete group $\Gamma$ with an irreducible unitary representation $ \rho$ and satisfying $u(\gamma z)=\rho(\gamma)u(z)$. The Lax-Phillips approach is used with the wave equation playing a central role. Incoming and outgoing subspaces are employed to obtain corresponding unitary translation representations, $R_-$ and $R_+$, for the solution operator. The scattering operator, which maps $R_-f$ into $R_+f$, is unitary and commutes with translation. The spectral representation of the scattering operator is a multiplicative operator, which can be expressed in terms of the constant term of the Eisenstein series. When the dimension of $\rho$ is one, the elements of the scattering operator cannot vanish. However when $\dim(\rho)>1$ this is no longer the case.

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Ralph Phillips
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305

Received by editor(s): January 23, 1996
Article copyright: © Copyright 1997 American Mathematical Society