# Microscopic Derivation of Time-dependent Point Interactions

## TL;DR

This paper derives a microscopic, time-dependent point interaction model for quantum particles in a semi-classical regime, providing a rigorous approximation of the dynamics of a polaron coupled to bosonic fields.

## Contribution

It introduces a novel derivation of time-dependent point interactions as effective models for quantum particles in intense field regimes.

## Key findings

- Effective dynamics approximated by time-dependent point interactions.
- Strong operator topology approximation of unitary dynamics.
- Connection between regular potentials and point interactions.

## Abstract

We study the dynamics of the three-dimensional polaron - a quantum particle coupled to bosonic fields - in the quasi-classical regime. In this case the fields are 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 by-product, 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 Schr\"{o}dinger operators with suitably rescaled regular potentials.