Spectral decomposition of pseudo-cuspforms, and meromorphic continuation of Eisenstein series, on $\mathbb{Q}$-rank one arithmetic quotients

TL;DR

This paper extends spectral theory results for pseudo-cuspforms and proves the meromorphic continuation of Eisenstein series on certain arithmetic quotients, using Friedrichs extension methods instead of semigroup approaches.

Contribution

It generalizes Lax-Phillips' theorem on pseudo-cuspforms and establishes meromorphic continuation of Eisenstein series via Friedrichs self-adjoint extensions.

Findings

Extended Lax-Phillips' theorem to new settings

Proved meromorphic continuation of Eisenstein series in several cases

Used Friedrichs extension approach instead of semigroup methods

Abstract

We extend Lax-Phillips' theorem on discreteness of pseudo-cuspforms, in the style of Colin de Verdi{\`e}re's use of the Friedrichs self-adjoint extension of a restriction of the Laplace-Beltrami operator, as opposed to the use of semigroup methods. We use this to prove meromorphic continuation of Eisenstein series in several $\mathbb{Q}$-rank one cases, again following Colin de Verdi{\`e}re, as opposed to the semigroup-oriented viewpoint of Lax-Phillips and W. Mueller.