Emergence of Time-Dependent Point Interactions in Polaron Models

We study the dynamics of the three-dimensional Fröhlich polaron - a quantum particle coupled to a bosonic field - in the quasi-classical regime, i.e., when the field is very intense and the corresponding degrees of freedom can be treated semiclassically. We prove that in such a regime the effective dynamics for the quantum particles is approximated by the one generated by a time-dependent point interaction, i.e., a singular time-dependent perturbation of the Laplacian supported in a point. As a byproduct, we also show that the unitary dynamics of a time-dependent point interaction can be approximated in strong operator topology by the one generated by time-dependent Schrödinger operators with suitably rescaled regular potentials.