I asked this question already on stackexchange, but I did not get any resonance at all, so maybe anybody here can give me a few hints about my problem.

My goal is to solve this PDE for $f:[-1,1] \times \mathbb{R}_{\ge 0}\rightarrow \mathbb{C}$ $$ i\partial_t f(x,t) = -\partial_x^2 f(x,t) + g(t)V(x)f(x,t).$$

I would consider this PDE to be solved if I get two ODEs just depending on either $x$ or $t$. $$f(x,0)$$ is specified a priori and $\int_{[-1,1]} f^*(x,t)f(x,t) dx=1$ for all $t \ge 0$.

Separation of variables seems to fail here and also integral transforms appear to be useless. Despite, I don't want to use perturbative techniques.

I want to have that $g$ is a $C^{\infty}$ function with compact support and $V \in C^{\infty}$.

A simpler setting where an integral transform could maybe work is this one:

If we take $g(t):=\delta(t-t_0)$, where $t_0>0$.

$$ \partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t-t_0)V(x)f(x,t).$$

On the other hand, the integral transform(especially the Fourier transform) seems to fail, as the function does not need to be square integrable with respect to time. So, I don't really see if we can do anything about it.