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Recently I asked the related question but this one is more complicated. If one has a hydrogen atom interacting with a laser pulse with the hamiltonian: $$H = \frac{1}{2}{\left[ {\hat p - \frac{1}{c}\vec A(\vec r,t')} \right]^2} - V(\vec r),$$ then knowing the ground state energy of a hydrogen atom one can write the solution as $$\psi = {\psi _0}(t = 0){e^{ - iE(t - {t_0})}}\exp \left[ { - \int\limits_{{t_0}}^t {\left( {\frac{i}{c}(\vec \nabla ,\vec A(\vec r,t')) - \frac{1}{{2{c^2}}}{{\vec A}^2}(\vec r,t')} \right)dt'} } \right]$$ Is it possible to integrate the term $\exp \left[ { - \int\limits_{{t_0}}^t {\left( {\frac{i}{c}(\vec \nabla ,\vec A(\vec r,t')) - \frac{1}{{2{c^2}}}{{\vec A}^2}(\vec r,t')} \right)dt'} } \right]?$

Or do I need to use a propagator routine? But if it is the case then can I begin with Gordon-Volkov state for a free electron in laser field like in this paper https://journals.aps.org/pra/abstract/10.1103/PhysRevA.8.804?

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