Hi

This is a general question about the possibilities to solve the time dependent Schrödinger equation:

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

So I am assuming that we have a time-dependent potential that consists of a function $g \in L^2 \cap C^{\infty}$ and a potential $V \in C^{\infty}$

I would consider this PDE to be solved if I get two ODEs just depending on either $x$ or $t$. ( Though separation of variables does not work directly a good substitution or integral transform could do it)

Furthermore, I want to specify the wavefunction $f(.,t_0)$ at a particular point in time $t_0$ as a boundary condition. The other boundary conditions shall be time-independent.

As an example, we could look at the PDE where $f(.,0)$ is given and at some point in time $t_0>0$ we apply a short potential pulse $V$ to the wavefunction.

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

Can we say anything in this general setting about ways to solve this PDE.

Of course, perturbation theory or any standard ways of approximations are not interesting to me.

If anything about my question is unclear, please let me know.