What is the spectral theorem for compact self-adjoint operators on a Hilbert space actually for? Please excuse the naive question.  I have had two classes now in which this theorem was taught and proven, but I have only ever seen a single (indirect?) application involving the quantum harmonic oscillator.  Even if this is not the strongest spectral theorem, it still seems useful enough that there should be many nice examples illustrating its utility.  So... what are some of those examples?  
(I couldn't readily find any nice examples looking through a few functional analysis textbooks, either.  Maybe I have the wrong books.)

The only way I know to prove "discreteness" of some piece of a spectrum is to find one or more compact operators on it, suitably separating points. That is, somehow the only tractable operators are those closely related to compact ones.
Even to discuss the spectral theory of self-adjoint differential operators $T$, the happiest cases are where $T$ has compact resolvent $(T-\lambda)^{-1}$.
In particular instances, the Schwartz kernel theorem depends on the compactness of the inclusions of Sobolev spaces into each other (Rellich's lemma). 
In automorphic forms: to prove the discreteness of spaces of cuspforms, one shows that the natural integral operators (after Selberg, Gelfand, Langlands et alia) restricted to the space of $L^2$ cuspforms are compact.
One of Selberg's arguments, Bernstein's sketch, Colin de Verdiere's proof, and (apparently) the proof in Moeglin-Waldspurger's book (credited to Jacquet, credited to Colin de Verdiere!?) of meromorphic continuation of Eisenstein series of various sorts depends ultimately on proving compactness of an operator.
