Separation of variables, when possible?

Question

$$ \Delta \Psi(x, y, z) + V(x, y, z)\Psi(x, y, z) = E \Psi(x, y, z) $$

For which $V(x, y, z)$ can this partial differential equation (eigenproblem) be solved by separation of variables?

Answer 1

Apply separation of variables

$$\Psi(x,y,z) = X(x)Y(y)Z(z)$$

and the equation becomes

$$\frac{d^2X}{Xdx^2} + \frac{d^2Y}{Ydy^2} + \frac{d^2Z}{Zdz^2} + V(x,y,z) = E$$

Now in order to be able to split this into three seperate equations we cannot have any coupling, i.e. cross-terms like $xy^2$, between any of the variables in $V$. Thus we must have

$$V(x,y,z) = A(x) + B(y) + C(z)$$

Disclaimer: For some equations you might be able to perform a change of variables $(x,y,z)\to(x',y',z')$ where we have cross-terms in $V(x,y,z)$ but not in $V(x',y',z')$. Now if the Laplacian in the new variables still remains simple in the sense that it does not mix derivatives of $(x',y',z')$ then seperation should also work here. How to know if this can be the case for a given equation (or if this is possible at all): I have no clue!