Bilinear estimate for Schr\"odinger equation on $\mathbb{R} \times \mathbb{T}$

TL;DR

This paper advances bilinear estimates for the Schrödinger equation on waveguides, extending previous results beyond semiclassical times and establishing sharp bounds with matching examples.

Contribution

It introduces sharper bilinear estimates on waveguides that surpass previous semiclassical limitations, with proofs of sharpness through explicit examples.

Findings

Established bilinear estimates beyond semiclassical time regime.

Proved the sharpness of the estimates with explicit constructions.

Extended the understanding of Schrödinger dynamics on waveguides.

Abstract

We continue our study of bilinear estimates on waveguide $\mathbb{R}\times \mathbb{T}$ started in \cite{DFYZZ2024,Deng2023}. The main point of the current article is, comparing to previous work \cite{Deng2023}, that we obtain estimates beyond the semiclassical time regime. Our estimate is sharp in the sense that one can construct examples which saturate this estimate.