TY - JOUR
T1 - Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group
AU - Bahouri , Hajer
AU - Gallagher , Isabelle
JO - Communications in Mathematical Research 
VL - 1
SP - 1
EP - 35
PY - 2022
DA - 2022/10
SN - 39
DO - http://doi.org/10.4208/cmr.2021-0101
UR - https://global-sci.org/intro/article_detail/cmr/21076.html
KW - Heisenberg group, Schrödinger equation, dispersive estimates, Strichartz estimates.
AB - <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>