Spectral Scattering Theory for Automorphic Forms

We construct a scattering process for L2-automorphic forms on the quotient of the upper half plane by a cofinite discrete subgroup Γ of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SL_{2}({\mathbb{R}})$$\end{document}. The construction is algebraic besides being analytic in the sense that we use some relations satisfied by real-analytic Eisenstein series with a complex parameter. Thanks to this feature, the construction of our operators and spaces is explicit. We show some properties of the Lax-Phillips generator on a scattering subspace carved out from this process. We prove that the spectrum of this operator consists only of eigenvalues, which correspond to the nontrivial zeros, counted with multiplicity, of the Dirichlet series appearing in the functional equation of the Eisenstein series. In particular, in the case of the (full) modular group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SL_{2}({\mathbb{Z}})$$\end{document}, the Dirichlet series reduces to the Riemann zeta function ζ, thereby we obtain a spectral interpretation of the nontrivial zeros of ζ.