Time decay for solutions of Schr\"odinger equations with rough and time dependent potentials

TL;DR

This paper proves dispersive and Strichartz estimates for solutions to linear Schrödinger equations with small, rough, and time-dependent potentials, advancing understanding of their behavior under less regular conditions.

Contribution

It establishes new dispersive and Strichartz estimates for Schrödinger equations with rough, time-dependent potentials, including quasiperiodic and small decay potentials, addressing previously open questions.

Findings

Dispersive estimates for small rough time-dependent potentials.

Strichartz estimates for potentials decaying faster than |x|^{-2}.

Extension of estimates to potentials in intersection of Rollnik and Kato classes.

Abstract

We establish dispersive and Strichartz estimates for solutions to the linear time-dependent Schr\"odinger equations with potential in three dimensions. Our main focus is on the small rough time-dependent potentials. Examples of such potentials are of the form $V(t,x)=T(t) V_0(x)$, where $T$ is quasiperiodic in time and $V_0$ is essentially an $L^{3/2}$ function of the spatial variables. We also prove the dispersive estimates for small time-independent potentials which belong to the interestion of the Rollnik and global Kato classes. Finally, we settle the question posed by Journe, Soffer, Sogge concerning Strichartz estimates for potentials that decay faster than $|x|^{-2}$.