Solution to Schrödinger equation 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.
 A: 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 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.

A: In the linear case, $V(x)=x$, the Schrodinger equation can be solved by Lewis-Riesenfeld approach. See http://link.springer.com/article/10.1007%2Fs12648-013-0322-4 as well as http://arxiv.org/abs/quant-ph/0309174 and references cited therein.
Some other cases is considered in http://journal.kcsnet.or.kr/main/j_search/j_abstract_view.htm?code=B021211&qpage=j_search&spage=b_bkcs&dpage=ar (Exactly Solvable Time-Dependent Problems: Potentials of Monotonously Decreasing Function of Time, by Tae Jun Park).
