Constraints on potential for normalizable wavefunction

[Gfunction]

We know that in one dimension if $E>V(\infty)$ or $E>V(-\infty)$ then the resulting wave function will not be normalizable. The basic argument is that if $E>V(\infty)$, then a stationary solution to the Schrödinger equation will necessarily have a concavity with the same sign as the solution itself for all $x$ greater than some value $a$. So if the resulting wavefunction was positive after $a$, then the wavefunction would also be concave up and curve away from the x-axis. Then the integral of the squared wavefunction would tend towards infinity since the wavefunction would never again go to $0$. A similar argument can be made if the wavefunction was negative after $a$ and again for $E>V(-\infty)$. We don't have this problem with the infinite square well or simple harmonic oscillator because the potentials go to infinity at $\pm \infty$.

In short, my question is this: What are necessary and sufficient conditions for a potential to generate normalizable solutions?

I would prefer to keep things simple with a one-dimensional treatment that doesn't need to be as rigorous as say a functional analysis proof (so maybe constrain ourselves to potentials that are at least C2). But after that's accomplished, I definitely wouldn't mind a more advanced discussion.

 

[azntoon]

The necessary and sufficient conditions for a potential to generate normalizable solutions are that the potential must be bounded from below, meaning that the potential must have a finite limit as x approaches infinity, and also that the potential must have an upper bound, meaning that the potential must have a finite limit as x approaches negative infinity. In addition, the potential must have properties that allow the wavefunction to fall off to zero at infinity, such as having a non-zero gradient at the boundaries, or having a periodicity that allows it to repeat in a way that it reaches zero at infinity. This is because the integral of the square of the wavefunction must be finite for the wavefunction to be normalizable.