Consider $u_t = i u_{xx} - x^2 u$ with $u_{t = 0} = 1$.
We want to find a solution.
My attempt : let's say $u = X(x)T(t)$, hence we have $\frac{T'(t)}{T(t)} = i \frac{X"(x)}{X(x)} - x^2$.
We may say that $\frac{T'(t)}{T(t)} = \lambda$ and $i \frac{X"(x)}{X(x)} - x^2 = \lambda$ , so $ \frac{X"(x)}{X(x)} = -i(x^2 + \lambda)$.
But the second equation is give some problem.
Maybe there is a better way to solve it?
UPD : I actually think that this idea is bad, because if $u = X(x)T(t)$, when $u(x,0) = X(x) = 1$