Lossless Strichartz estimates on rectangular tori over short time intervals

TL;DR

This paper establishes sharp Strichartz estimates for the Schrödinger equation on rectangular tori over short time intervals, improving understanding of dispersive properties in this setting.

Contribution

It proves lossless Strichartz estimates at critical and endpoint exponents for the Schrödinger equation on rectangular tori with frequency-localized data over small time windows.

Findings

Established lossless Strichartz estimates at critical exponent $q_c$

Proved endpoint estimates for specific exponent pairs

Results hold for high-frequency localized initial data

Abstract

We prove lossless Strichartz estimates at the critical exponent $q_c = \frac{2(n+1)}{n-1}$ and the endpoint exponent pair $\left(2,\frac{2(n-1)}{n-3}\right)$ for the Schr\"{o}dinger equation on rectangular tori of dimension $n-1$ with frequency localized initial data on small time windows with length depending on the frequency parameter $\lambda \gg 1$.