TITLE: Harmonic oscillator discrete spectrum
QUESTION [8 upvotes]: Let us act intentionally stupid and assume we do not know that we can solve for the spectrum of the harmonic oscillator 
$$-\frac{d^2}{dx^2}+x^2$$
explicitly.
Is there an abstract argument why the spectrum of this operator is discrete and tends to infinity?
It almost looks like a result that would follow from the compactness of some operator involved, yet we are here on an unbounded domain $\mathbb{R}.$
So does it follow from spectral theory only that the spectrum is discrete?

REPLY [4 votes]: The physical idea is that $-\Delta+q$ for a sufficiently-growing "confining potential" $q$ on $\mathbb R^n$ should have compact resolvent, which then proves discreteness of its spectrum. In many explicit situations, one can prove this directly by imitating proofs of Rellich's compactness lemma(s).
Not an abstract version, but maybe non-trivial, is illustrated for automorphic forms in http://www.math.umn.edu/~garrett/m/v/simplest_afc_schrodinger.pdf