Time-dependent delta-interactions for 1D Schr\"odinger Hamiltonians

TL;DR

This paper investigates time-dependent point interactions in 1D Schrödinger equations, revealing that existing regularity assumptions are overly restrictive and introducing novel applications of paraproduct techniques to linear quantum problems.

Contribution

It provides refined regularity conditions for time-dependent delta interactions and applies paraproduct methods from nonlinear PDEs to a linear quantum framework.

Findings

Regularity assumptions for coupling parameters are improved.

Paraproduct techniques are successfully applied to linear Schrödinger problems.

The results challenge existing notions of optimality in non autonomous evolution equations.

Abstract

The non autonomous Cauchy problem for time dependent 1D point interactions is considered. The regularity assumptions for the coupling parameter are accurately analyzed and show that the general results for non autonomous linear evolution equations in Banach spaces are far from being optimal. In the mean time, this article shows an unexpected application of paraproduct techniques, initiated by J.M. Bony for nonlinear partial differential equations, to a classical linear problem.