In Griffiths' text on QM, I am trying to understand his logic as to why there can be no states of negative energy. He writes:
What if I apply the lowering operator repeatedly? Eventually I'm going to reach a state with energy less than zero, which (according to the general theorem of Problem 2.2) does not exist!
And here is Problem 2.2:
Show that $E$ must exceed the minimum value of $V(x)$ for every normalizable solution to the time-independent Shroedinger equation.
Taking this as a given, is the reason we can't have negative energy states for the quantum harmonic oscillator because, in this situation, the potential is always non-negative? Otherwise, I don't see how Problem 2.2 helps us.
More generally, if the potential of a system is negative, then it is possible to have negative energy states?