So I was thinking today about when observables become "quantized", and came to the conclusion that every instance of quantization I've ever come across has come about from solving the Schrödinger equation over a compact region.
This seemed to hold for everything I could think of which exception to the radial solution to the hydrogen atom. The region over which it is solved is $r\in [0, \infty)$ (the angular regions are compact). Also, the Simple Harmonic Oscillator is an example. (I'm sure there are more)
Question: Is there a more general notion of "quantization" that will work in all cases? That is, a bidirectional "if and only if" statement. And if not, why isn't that a huge problem with our theory?