The Schrödinger's equation can be written in this form:
$-u''(x)+V(x) u(x) = E u(x) $
$V(x)$ is a function that is defined on the real line. We know ${u}^{2}$ is integrable on the whole real line. We are interested in determining $E$. Here is my question: under what condition(s) on $V$ and/or $u$, allowed numbers for $E$ are denumerable? (As I have studied in quantum mechanics, we are expected eigen-energies to be denumerable in physical(?) situations.)