Potential such that the time-independent Schroedinger equation has an explicit solution

Question

Consider the time-independent Schroedinger equation

$\phi'' (x) +V(x)\phi(x)=\lambda \phi(x)$, $\quad x\in\mathbb{R}$.

For testing my numerics, I would like to now:

For which choices of $V$ are there explicit expressions for the solutions $(\phi,\lambda)$?

For example, if $V(x)=x^2$, then the solutions are products of Hermite polynomials and $exp(-x^2)$ (up to correct scales) and arithmetic progressions (the same works in higher dimensions with the analogue potential).

Answer 1

You can always work backwards if you just want a single solution: start with known physically plausible choices of any $\phi(x)$ and $\lambda$, then insert these into the Schrodinger equation to determine the $V(x)$. (Caveat. This is a math hack for getting a single explicit solution that can be tested numerically, not a physically reasonable approach to finding ALL solutions.)