Quantum Coulomb problem in a Gaussian time-dependent electric field within the path integral formalism

We solve the Coulomb problem in nonrelativistic quantum mechanics with a charge depending on a parameter that plays the role of time. The choice of this dependence is needed, for example, after certain spatio–temporal transformations when dealing with the interaction of a “small” system (quantum sub-system) with a “large” system, e.g., a bath. These spatio–temporal transformations, combined with path integral, allow us to find the Feynman propagator of the quantum subsystem. To test our approach, we derive the pure Coulomb Green's function as a limit of our result.