Two-dimensional Time-dependent Point Interactions

TL;DR

This paper investigates the quantum dynamics of particles under time-dependent point interactions in two dimensions, extending known results from other dimensions by deriving a system of equations that fully describes wave evolution.

Contribution

It introduces a novel framework for analyzing two-dimensional time-dependent point interactions using charge equations, expanding the theoretical understanding beyond one and three dimensions.

Findings

Wave packet evolution is determined by solutions to Volterra-type charge equations.

The approach generalizes existing models to two-dimensional systems.

Provides a mathematical foundation for future studies of quantum particles with point interactions.

Abstract

We study the time-evolution of a quantum particle subjected to time-dependent zero-range forces in two dimensions. After establishing a conceivable ansatz for the solution to the Schr\"{o}dinger equation, we prove that the wave packet time-evolution is completely specified by the solutions of a system of Volterra-type equations -- the {\it charge equations} -- involving the coefficients of the singular part of the wave function, thus extending to the two-dimensional case known results in one and three dimensions.