The Lax–Phillips infinitesimal generator and the scattering matrix for automorphic functions
Volume 92 / 2007
Abstract
We study the infinitesimal generator of the Lax–Phillips semigroup of the automorphic scattering system defined on the Poincaré upper half-plane for ${\rm SL}_2(\mathbb{Z})$. We show that its spectrum consists only of the poles of the resolvent of the generator, and coincides with the poles of the scattering matrix, counted with multiplicities. Using this we construct an operator whose eigenvalues, counted with algebraic multiplicities (i.e. dimensions of generalized eigenspaces), are precisely the non-trivial zeros of the Riemann zeta function. We give an operator model on $L^2(\mathbb{R})$ of this generator as explicit as possible. We obtain a condition equivalent to the Riemann hypothesis in terms of cyclic vectors for a weak resolvent of the scattering matrix.
