Local dispersive and Strichartz estimates for the Schr{\"o}dinger operator on the Heisenberg group

TL;DR

This paper develops local dispersive and Strichartz estimates for the Schr{"o}dinger equation on the Heisenberg group, overcoming the non-dispersive nature of the global evolution by analyzing the Schr{"o}dinger kernel with advanced harmonic analysis techniques.

Contribution

It introduces refined local dispersive estimates for the Schr{"o}dinger operator on the Heisenberg group, utilizing explicit heat kernel formulas and complex analysis methods.

Findings

Establishment of sharp local dispersive estimates on b^d

Derivation of local Strichartz estimates for the Schrd6dinger equation

Identification of kernel concentration on quantized hyperplanes

Abstract

It was proved by H. Bahouri, P. G{\'e}rard and C.-J. Xu in [9] that the Schr{\"o}dinger equation on the Heisenberg group $\mathbb{H}^d$, involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on $\mathbb{H}^d$ for the linear Schr{\"o}dinger equation, by a refined study of the Schr{\"o}dinger kernel $S_t$ on $\mathbb{H}^d$. The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on $\mathbb{H}^d$ derived by B. Gaveau in [20], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results, we establish local Strichartz estimates and prove that the kernel $S_t$ concentrates on quantized horizontal hyperplanes of $\mathbb{H}^d$.