I was studying Schrödinger's equation for a school paper and wanted to solve it in a bit broader context than the potential well. I wanted to use a potential function $$V(x,y,z) = \chi(x) + \gamma(y) + \zeta(z)$$ for continuous (maybe continuously differentiable, don't know if that will matter later) functions $\chi(x), \gamma(y)$, and $\zeta(z)$. After using separation of variables, I get this equation (and others analogous for y and z): $$X''(x) + (\chi(x) + \alpha)X(x) = 0$$ Where alpha is proportional to the separation constant. I was wondering whether there is a way to solve this without directly specifying $\chi(x)$. It wouldn't be a problem to have series and/or integrals.
Should that not be possible (as is my hunch), is there a nice solution if I let $\chi(x)$, $\gamma(y)$, and $\zeta(z)$ be polynomials? I heard this may be possible using something called "Hermite polynomials", which I'm far from familiar with, so would appreciate some help here as well!
Thanks!