TITLE: Changing the manifold, preserving the discrete spectrum
QUESTION [5 upvotes]: On a Riemannian manifold $M$, the Laplace operator $L$ is uniquely defined. 
If $M$ is not compact, then $L$ admits a continuous spectrum.
Is there a way of "changing" $M$ and/or $L$ in say $M'$ and $L'$, such that the spectrum of $L'$ is now purely discrete and contains the spectrum of $L$? I am interested in examples of this type.
Feel free to restrict to homogenous varieties or any other geometric gadget, if have an example of similar nature in mind.
Motivation: More general it is true that, given an self adjoint operator $T$ on a Hilbert space, does there exists a decomposition such that $T_1 + T_2 = T$, where $T_1$ has the same point spectrum as $T$. I am not looking for this. Is there a better geometric construction?

REPLY [2 votes]: A different "adjustment" is as in Colin-de-Verdiere's 1981 treatment of meromorphic continuation of Eisenstein series (at least for rank-one?): he replaces the usual Laplacian (which is essentially self-adjoint) by a pseudo-Laplacian created as the minimal ("Friedrichs") extension of the restriction of the Laplacian to automorphic forms whose constant terms vanish above a fixed height. A variant of the argument for discreteness of cuspforms proves that this pseudo-Laplacian has a compact resolvent, hence, discrete spectrum. The cuspforms remain eigenfunctions, the constants cease to be eigenfunctions because they are not in the space, and certain truncated Eisenstein series become genuine eigenfunctions for the pseudo-Laplacian (rather than failing to be eigenfunctions for the ordinary Laplacian). 
Edit: and, in regard to some aspects of @Kimball's points, specifically (looking at the details of C-de-V's discussion), the truncated Eisenstein series are essentially those whose truncated constant term is still continuous, that is, so that, at height $y_o$, in the simplest case the condition for the truncation $\wedge^{y_o} E_s$ to be a "new" eigenfunction is that $y_o^s+c_s y_o^{1-s}=0$, where $c_s=\xi(2s-1)/\xi(2s)$. Thus, standard (if non-trivial) facts about zeta (and a refined Weyl's Law for cuspforms?) still assure that the "new" eigenfunctions are asymptotically fewer, quantifiably.