Loading [MathJax]/jax/output/HTML-CSS/config.js
full
search.png

Search

close.png

Cancel

arrow
Volume 39, Issue 1
Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group

Hajer Bahouri & Isabelle Gallagher

DOI: 10.4208/cmr.2021-0101

Commun. Math. Res., 39 (2023), pp. 1-35.

Published online: 2022-10

Preview Full PDF 3121 34693

Cited by

google scholar semantic scholar
Export citation
  • 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 $\mathbb{H}^d.$

  • Keywords

Heisenberg group, Schrödinger equation, dispersive estimates, Strichartz estimates.

  • AMS Subject Headings

43A30, 43A80

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{CMR-39-1, author = {Bahouri , Hajer and Gallagher , Isabelle}, title = {Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group}, journal = {Communications in Mathematical Research }, year = {2022}, volume = {39}, number = {1}, pages = {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 $\mathbb{H}^d.$

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2021-0101}, url = {http://global-sci.org/intro/article_detail/cmr/21076.html} }
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 -

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 $\mathbb{H}^d.$

Bahouri , Hajer and Gallagher , Isabelle. (2022). Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group. Communications in Mathematical Research . 39 (1). 1-35. doi:10.4208/cmr.2021-0101
Copy to clipboard
BibteX RIS TXT
The citation has been copied to your clipboard