I'm trying to solve a 1D time-dependent Schrodinger equation:
$$ i\,\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)\,x\right]\!\psi(x,t) $$
where $V(x)$ is a periodic Dirac delta potential (or the Dirac comb):
$$ V(x)=\sum_{k=-\infty}^\infty\delta(x-ka) $$
and $a$ is some constant.
The driving force $F(t)$ is some arbitrary function of time, but for simplicity can be assumed to be simple trigonometric functions here:
$$ F(t)=F_0\cos(\omega_0t) $$
where $F_0$ and $\omega_0$ are constants ($F_0$ is not weak so that the perturbation theory may not apply).
Or for case of potential is a simple trigonometric function
$$ V(x)=\cos(a x). $$
How can I solve this 1D TDSE analytically?