Time dependent delta-prime interactions in dimension one

TL;DR

This paper analyzes the Schrödinger equation with a time-dependent delta-prime interaction in one dimension, proving existence of solutions under weaker regularity conditions on the interaction strength than previously known.

Contribution

It establishes the existence of strong solutions for the Schrödinger equation with a time-dependent delta-prime interaction under fractional Sobolev regularity assumptions on the interaction strength.

Findings

Strong solutions exist if the interaction strength varies in $H^{3/4}( )$.

Solution expressed via free evolution and a Volterra integral equation.

Weakens regularity conditions compared to previous results.

Abstract

We solve the Cauchy problem for the Schr\"odinger equation corresponding to the family of Hamiltonians $H_{\gamma(t)}$ in $L^{2}(\mathbb{R})$ which describes a $\delta'$-interaction with time-dependent strength $1/\gamma(t)$. We prove that the strong solution of such a Cauchy problem exits whenever the map $t\mapsto\gamma(t)$ belongs to the fractional Sobolev space $H^{3/4}(\mathbb{R})$, thus weakening the hypotheses which would be required by the known general abstract results. The solution is expressed in terms of the free evolution and the solution of a Volterra integral equation.