Smoothing And Dispersive Estimates For 1d Schr\"odinger Equations With BV Coefficients And Applications

TL;DR

This paper establishes smoothing and dispersive estimates for 1D Schrödinger equations with BV coefficients, matching constant coefficient results, and applies these findings to well-posedness of a generalized Benjamin-Ono equation.

Contribution

It proves optimal dispersive and Strichartz estimates for Schrödinger equations with BV coefficients, extending classical results to variable coefficient settings.

Findings

Smoothing estimates hold for BV coefficient Schrödinger equations.

Optimal Strichartz and maximal function estimates are achieved, matching constant coefficient cases.

Counterexamples show BV regularity is minimal for these estimates.

Abstract

We prove smoothing estimates for Schr\"odinger equations $i\partial_t \phi+\partial_x (a(x) \partial_x \phi) =0$ with $a(x)\in \mathrm{BV}$, the space of functions with bounded total variation, real, positive and bounded from below. We then bootstrap these estimates to obtain optimal Strichartz and maximal function estimates, all of which turn out to be identical to the constant coefficient case. We also provide counterexamples showing $a\in \mathrm{BV}$ to be a minimal requirement. Finally, we provide an application to sharp wellposedness for a generalized Benjamin-Ono equation.