I want to solve $$ i\partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t-t_0)V(x)f(x,t),$$
for any $V \in C^{\infty}[-1,1]$ and $f: [-1,1] \times \mathbb{R_{\ge 0}} \rightarrow \mathbb{C}$.
I would consider this to be solved if I have one/two ODEs that just depend on either $x$ or $t$ (or an integral transform etc.). My boundary conditions shall be taken such, that I specify a $f(.,0)$ for some $t_0 \in \mathbb{R}$. and look how this solution propagates under this ODE. The other boundary conditions shall be time-independent.
I also noticed that this equation is similar to many diffusion equations and found a similar differential operator here at http://en.wikipedia.org/wiki/Green%27s_function#Table_of_Green.27s_functions
This question is a particular example of my unsolved question here: https://math.stackexchange.com/questions/843106/solve-pde-by-getting-two-odes