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

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