In standard PDE theory, one generates eigenvalues to Sturm-Liouville problems over a finite domain. So, for a wave equation, we have an infinite number of eigenvalues $_$ for a Dirichlet problem, say.
What is the mathematical justification for having an infinite number of energy eigenvalues for the quantum harmonic oscillator? The eigenfunctions are Hermite functions, but over the infinite domain, $-\infty < x < \infty$. Is there some existence theorem that guarantees this discrete set of eigenvalues even on an infinite domain?