The Schroedinger Equation with Potential in Rough Motion

TL;DR

This paper establishes endpoint Strichartz estimates for the Schrödinger equation with a rough, time-dependent potential in three dimensions, and explores implications for bound states and energy boundedness, including a nonlinear application.

Contribution

It proves endpoint Strichartz estimates for Schrödinger equations with non-smooth, large, time-dependent potentials and analyzes their effects on bound states and energy.

Findings

Proved endpoint Strichartz estimates for rough, time-dependent potentials.

Demonstrated non-dispersion of bound states under small path perturbations.

Established boundedness of energy for the system.

Abstract

This paper proves endpoint Strichartz estimates for the linear Schroedinger equation in $R^3$, with a time-dependent potential that keeps a constant profile and is subject to a rough motion, which need not be differentiable and may be large in norm. The potential is also subjected to a time-dependent rescaling, with a non-differentiable dilation parameter. We use the Strichartz estimates to prove the non-dispersion of bound states, when the path is small in norm, as well as boundedness of energy. We also include a sample nonlinear application of the linear results.