The Lax–Phillips infinitesimal generator and the scattering matrix for automorphic functions

Yoichi Uetake¹

¹ Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Poznań, Poland

Annales Polonici Mathematici, Tome 92 (2007) no. 2, pp. 99-122.

Résumé

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.

Détail

DOI : 10.4064/ap92-2-1

Keywords: study infinitesimal generator lax phillips semigroup automorphic scattering system defined poincar upper half plane mathbb its spectrum consists only poles resolvent generator coincides poles scattering matrix counted multiplicities using construct operator whose eigenvalues counted algebraic multiplicities dimensions generalized eigenspaces precisely non trivial zeros riemann zeta function operator model mathbb generator explicit possible obtain condition equivalent riemann hypothesis terms cyclic vectors weak resolvent scattering matrix