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In the context of spectral decomposition of functions in $L^2(\Gamma \backslash \mathfrak{h})$, or Selberg trace formula, we come across three different types of spectrum.

First off there is the cuspidal spectrum, which consists of the eigenfunctions of the Laplacian $\Delta_k$ which also vanish at all the cusps. Secondly there is the continuous spectrum, which is spanned by the Eisenstein series which consists of $E(z,\frac 12 + it)$. Finally there is the residual spectrum which consists of functions $\operatorname{Res}_{s_0}(z,s)$ for various $s_0$.

From the point of view of functional analysis of the operator $\Delta_k$ what distinguishes the residual spectrum from the cuspidal spectrum? They both lie in $L^2$ and they both are eigenfunctions of the Laplacian.

In the terminology of functional analysis, one calls a number $\lambda$ in the residual spectrum of an operator $T$, if $T - \lambda I$ is injective but not surjective. I suspect that these two concepts of residuality have nothing to do with each other.

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Right, the two senses of "residual" are essentially unrelated. The distinctions between cuspidal discrete spectrum and other parts ("residual") of the discrete spectrum are not (so far as I know) at the completely abstract spectral level, but are about the explicit construction or classification: cuspforms versus square-integrable residues of various Eisenstein series. – paul garrett Mar 27 '12 at 17:00

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