Can this kind of TDSE be solved by series?
Note: some notation was changed according to the comments, after the first answer was posted.
I have a particular kind of time dependent <PERSON> equation:
$$i \hbar \frac{\partial}{\partial t} \Psi (t) =(\hat{H_0}+ \frac{t}{\tau} \hat{H_1}) \Psi (t) \ \Psi (0) = \Psi_0 $$
$\tau$ is some suitable time scale. The initial state is normalized $\langle \Psi_0 | \Psi_0 \rangle=1$.
In my case with appropriate coordinate transformation $$\hat{H_1} = i\frac{\partial}{\partial x}$$. 955
So the full Hamiltonian appears to be Hermitian.
Let's assume the wavefunction can be expressed as <PERSON> series in time (I don't see why not).
$$\Psi(t)=\sum_{n=0}^\infty C_n \frac{t^n}{\tau^n} \ C_0 = \Psi_0$$
Then by direct substitution we obtain:
$$i \frac{\hbar}{\tau} \sum_{n=0}^\infty C_{n+1} (n+1) \frac{t^n}{\tau^n} =\hat{H_0} \sum_{n=0}^\infty C_n \frac{t^n}{\tau^n} + \hat{H_1} \sum_{n=1}^\infty C_{n-1} \frac{t^n}{\tau^n}$$
Collecting the terms, we obtain the recurrence relation:
$$i \frac{\hbar}{\tau} (n+1) C_{n+1} = \hat{H_0} C_n + \hat{H_1} C_{n-1}, \qquad n \geq 1 \ i \frac{\hbar}{\tau} C_1= \hat{H_0} C_0$$
This seems like a full solution, assuming we can apply the operators $\hat{H}_{0,1}$ as many times as we want.
However, I have two questions:
* Is there any problem with my derivation? In general $[\hat{H}_0, \hat{H}_1] \neq 0$, could this lead to any issues?
* What about normalization? Would the series preserve the normalization of the initial state? If not, how do I fix that?
I know this is not the usual method for solving the TDSE, but I wanted to give it a try, since for my particular problem is seems the most suitable, as opposed, for example, to numerical scheme.
Edit:
About the explicit form of $\hat{H}_0$: I'm mostly interested in the case of exciton equation with confinement in one dimension, so it would look something like:
$$\hat{H}_0=- \frac{\partial^2}{\partial X^2}- \frac{\partial^2}{\partial x^2}- \frac{\partial^2}{\partial \rho^2}- \frac{1}{\rho}\frac{\partial}{\partial \rho}+U(X,x)-\frac{1}{\sqrt{x^2+\rho^2}}$$
Where I omitted all the coefficients for brevity. $X$ is the center of mass coordinate in one direction, $x$ is the distance between electron and hole in the same direction, $\rho$ is the distance in plane.
Update 2:
For what it's worth, I tried some numerical experiments in different representations, and it doesn't seem like this series converges at all. Maybe for some potentials it could be possible, but not for my case. 955