@Article{CMR-39-1,
author = {Hajer, Bahouri and Isabelle, Gallagher},
title = {Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group},
journal = {Communications in Mathematical Research },
year = {2023},
volume = {39},
number = {1},
pages = {1--35},
abstract = {<p style="text-align: justify;">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 Schr<span style="text-align: justify;">ö</span>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 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 $\mathbb{H}^d.$</p>},
issn = {2707-8523},
doi = {https://doi.org/10.4208/cmr.2021-0101},
url = {https://global-sci.com/article/81448/local-dispersive-and-strichartz-estimates-for-the-schrodinger-operator-on-the-heisenberg-group}
}