Year: 2023
Author: Hajer Bahouri, Isabelle Gallagher
Communications in Mathematical Research , Vol. 39 (2023), Iss. 1 : pp. 1–35
Abstract
It was proved by Bahouri et al. [9] that the Schrödinger 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 Schrödinger equation, by a refined study of the Schrödinger 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 Gaveau [19], 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 $\mathbb{H}^d.$
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cmr.2021-0101
Communications in Mathematical Research , Vol. 39 (2023), Iss. 1 : pp. 1–35
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 35
Keywords: Heisenberg group Schrödinger equation dispersive estimates Strichartz estimates.