Some remarks on the Schr\"odinger equation with a potential in $L^{r}_{t}L^{s}_{x}$

TL;DR

This paper investigates the dispersive properties of the linear Schrödinger equation with time-dependent potentials in $L^{r}_{t}L^{s}_{x}$ spaces, establishing conditions for Strichartz estimates and demonstrating their optimality through counterexamples.

Contribution

It provides the first comprehensive analysis linking $L^{r}_{t}L^{s}_{x}$ integrability of potentials to Strichartz estimates, including optimality results with counterexamples.

Findings

$L^{r}_{t}L^{s}_{x}$ integrability ensures Strichartz estimates for Schrödinger with potential.

Counterexamples show the sharpness of the integrability conditions.

Results apply to both local and global estimates for large potentials.

Abstract

We study the dispersive properties of the linear Schr\"odinger equation with a time-dependent potential $V(t,x)$. We show that an appropriate integrability condition in space and time on $V$, i.e. the boundedness of a suitable $L^{r}_{t}L^{s}_{x}$ norm, is sufficient to prove the full set of Strichartz estimates. We also construct several counterexamples which show that our assumptions are optimal, both for local and for global Strichartz estimates, in the class of large unsigned potentials $V\in L^{r}_tL^{s}_x$.