Consider the nonrelativistic quantum mechanics of one particle in one dimension ("NRQMOPOD") with the time-independent Schrodinger equation
$$ \left( -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x) \right) \psi(x) = E\ \psi(x), $$ where we assume that the potential energy function $V(x)$ is continuous and at large distances approaches a constant value $$ V_\infty = \lim_{|x| \to \infty} V(x) $$ in the extended real line (which is often set to 0 if $V_\infty$ is finite). We do not assume that the eigenfunction $\psi(x)$ is necessarily normalizable.
"Folk wisdom" says that the following three statements are equivalent:
- $\psi(x)$ is normalizable, i.e. $$\int \limits_{-\infty}^\infty dx\ |\psi(x)|^2 < \infty.$$ Or equivalently, $\psi(x)$ represents a bound state, i.e. $$\lim_{R \to \infty} \int \limits_{|x| > R} dx\ |\psi(x)|^2 = 0.$$ (The equivalence of these two statements is a straightforward exercise in real analysis.)
- $E < V_\infty$, and
- $E$ lies in a discrete part of the energy spectrum, i.e. there exists a proper energy interval such that $E$ is the only eigenvalue in the Hamiltonian's spectrum that lies within that interval.
But this folk wisdom is incorrect. This answer gives an explicit example of a potential energy function $V(x)$ and a normalizable energy eigenstate $\psi(x)$ with energy $E > V_\infty$. Therefore, statement #1 above does not imply statement #2. (I do not know whether the spectrum for the particular Hamiltonian given in that example is discrete or continuous around the relevant eigenvalue $\lambda = 1$, so I don't know the status of claim #3 for this example.)
What are the exact implications between the three statements above? Of the six possible implications, which have been proven to be true, which have explicit known counterexamples, and which are still open problems?
I'd also like to know about the case of multiple spatial dimensions, although I assume that the answers are probably the same as for the 1D case.
