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 Schrodinger 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.