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.



Solution to Schrödinger equation
I agree with @Christian Remling that the product structure of your potential $V(x,t)=g(t)V(x)$ is not helpful in general, but it would help if $g(t)$ is a monotonically decreasing function of time, see [this paper](https://web.archive.org/web/20160313020436/http://pdf.easechem.com/pdf/32/d34de190-5a35-4f5e-83d0-51d030387708.pdf) by T.J. Park (2002).



> 
> Only a few
> time-dependent systems have been reported to be analytically solved
> whose potentials are constant, linear, and quadratic functions of
> coordinate with arbitrary time-dependences. Here we show that time-dependent potentials of any
> coordinate-dependence can be solved exactly if the
> time-dependence is monotonously decreasing. We do this by a unitary
> transformation of the wavefunction and variable transformations to
> change the Schrödinger equation to be time-independent in new
> variables. These variables are then determined by solving a set of
> simple differential equations.
> 
> 
> 


