What can be said about the spectrum of the Laplace-Beltrami operator on a non-compact, complete Riemannian manifold of **finite** volume? For example, is the point spectrum non-empty?


What would be a good reference for this? 



Spectrum of Laplacian in non-compact manifolds
As far as I know, this is a very delicate question. That is, already in two dimensions, there can be only finite discrete spectrum. (See Phillips-Sarnak and Wolpert.)


Even on the modular curve $SL(2,\mathbb Z)\backslash \mathfrak H$, it was highly non-trivial to prove existence of infinitely-many $L^2$ eigenvalues, apparently requiring Selberg's invention and application of his "trace formula" in the 1950's.


Similarly, it has only been after 2000 or so that various people (S. Miller, Lapid-Finis, and others) have proven "Weyl's Law" for higher-rank quotients of semi-simple Lie groups, that is, that the asymptotics for discrete (even cuspidal) spectrum are what one would expect for a compact Riemannian manifold.


That is, based on this perspective, I'd be amazed if much could be said in general... maybe some negative results for small deformations, analogous to Phillips-Sarnak and Wolpert?


