Scattering Theory for Twisted Automorphic Functions

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 Γ with an irreducible unitary representation ρ and satisfying u(γz) = ρ(γ)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 ρ is one, the elements of the scattering operator cannot vanish. However when dim(ρ) > 1 this is no longer the case.