Energy eigenstates of a 1-dimensional particle are given by solutions to differential equations of the form $$ \left(-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x) \right) \psi(x) = E\psi(x) $$ where $V$ is the potential, $\psi$ is the energy eigenstate, and $E$ is the energy eigenvalue.
In introductory (and even fairly advanced) texts on quantum mechanics, the following facts are asserted without proof:
- The set $\{E \in \mathbb{R} : E \text{ is an energy eigenvalue and } E < \sup_{x \in \mathbb{R}} V(x) \}$ is a discrete subset of the reals bounded below by $\inf_{x \in \mathbb{R}} V(x)$.
- If the discrete energy eigenvalues above are listed in ascending order as $E_0, \ldots, E_n$ and have corresponding energy eigenstates $\psi_0, \ldots, \psi_n$, then $\psi_n$ will have $n + 1$ local maxima.
The above results are often stated with less precision, and sometimes even conflict with each other.
What statements like 1 and 2 above are actually true, and how does one go about proving them?