An Uncertainty Principle for Solutions of the Schrödinger Equation on H-Type Groups

In this paper we consider uncertainty principles for solutions of certain partial differential equations on H -type groups. We first prove that, on H -type groups, the heat kernel is an average of Gaussians in the central variable, so that it does not satisfy a certain reformulation of Hardy’s uncer...

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Authors: Fernández Bertolin, Aingeru, Jaming, Philippe, Pérez-Esteva, Salvador
Format: article
Publication Date:2020
Country:España
Institution:Universidad del País Vasco
Repository:Addi. Archivo Digital para la Docencia y la Investigación
OAI Identifier:oai:addi.ehu.eus:10810/52480
Online Access:http://hdl.handle.net/10810/52480
Access Level:Open access
Keyword:uncertainty principle
H-type group
Schrödinger equation
heat kernel
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spelling An Uncertainty Principle for Solutions of the Schrödinger Equation on H-Type GroupsFernández Bertolin, AingeruJaming, PhilippePérez-Esteva, Salvadoruncertainty principleH-type groupSchrödinger equationheat kernelIn this paper we consider uncertainty principles for solutions of certain partial differential equations on H -type groups. We first prove that, on H -type groups, the heat kernel is an average of Gaussians in the central variable, so that it does not satisfy a certain reformulation of Hardy’s uncertainty principle. We then prove the analogue of Hardy’s uncertainty principle for solutions of the Schrödinger equation with potential on H -type groups. This extends the free case considered by Ben Saïd et al. [‘Uniqueness of solutions to Schrödinger equations on H-type groups’, J. Aust. Math. Soc. (3) 95 (2013), 297–314] and by Ludwig and Müller [‘Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups’, Proc. Amer. Math. Soc. 142 (2014), 2101–2118].This study has been carried out with financial support from the French state, managed by the French National Research Agency (ANR) in the frame of the ‘Investments for the Future’ Programme IdEx Bordeaux—CPU (ANR-10-IDEX-03-02). A.F.-B. acknowledges financial support from ERCEA Advanced Grant 2014 669689—HADE, the MINECO project MTM2014-53145-P and the Basque Government project IT-641-13. P.J. acknowledges financial support from the French ANR program ANR-12-BS01-0001 (Aventures) from the Austrian–French AMADEUS project 35598VB—ChargeDisq and from the Tunisian–French CMCU/Utique project 15G1504. S.P.E. acknowledges financial support from the Mexican Grant PAPIIT-UNAM IN106418.Cambridge University PressEuropean Commission202120212020info:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10810/52480reponame:Addi. Archivo Digital para la Docencia y la Investigacióninstname:Universidad del País VascoInglésinfo:eu-repo/grantAgreement/EC/H2020/669689info:eu-repo/grantAgreement/MINECO/MTM2014-53145-P/https://doi.org/10.1017/S1446788720000026info:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by-nc-nd/3.0/es/CC BY-NC-NDoai:addi.ehu.eus:10810/524802026-06-18T09:23:17Z
dc.title.none.fl_str_mv An Uncertainty Principle for Solutions of the Schrödinger Equation on H-Type Groups
title An Uncertainty Principle for Solutions of the Schrödinger Equation on H-Type Groups
spellingShingle An Uncertainty Principle for Solutions of the Schrödinger Equation on H-Type Groups
Fernández Bertolin, Aingeru
uncertainty principle
H-type group
Schrödinger equation
heat kernel
title_short An Uncertainty Principle for Solutions of the Schrödinger Equation on H-Type Groups
title_full An Uncertainty Principle for Solutions of the Schrödinger Equation on H-Type Groups
title_fullStr An Uncertainty Principle for Solutions of the Schrödinger Equation on H-Type Groups
title_full_unstemmed An Uncertainty Principle for Solutions of the Schrödinger Equation on H-Type Groups
title_sort An Uncertainty Principle for Solutions of the Schrödinger Equation on H-Type Groups
dc.creator.none.fl_str_mv Fernández Bertolin, Aingeru
Jaming, Philippe
Pérez-Esteva, Salvador
author Fernández Bertolin, Aingeru
author_facet Fernández Bertolin, Aingeru
Jaming, Philippe
Pérez-Esteva, Salvador
author_role author
author2 Jaming, Philippe
Pérez-Esteva, Salvador
author2_role author
author
dc.contributor.none.fl_str_mv European Commission
dc.subject.none.fl_str_mv uncertainty principle
H-type group
Schrödinger equation
heat kernel
topic uncertainty principle
H-type group
Schrödinger equation
heat kernel
description In this paper we consider uncertainty principles for solutions of certain partial differential equations on H -type groups. We first prove that, on H -type groups, the heat kernel is an average of Gaussians in the central variable, so that it does not satisfy a certain reformulation of Hardy’s uncertainty principle. We then prove the analogue of Hardy’s uncertainty principle for solutions of the Schrödinger equation with potential on H -type groups. This extends the free case considered by Ben Saïd et al. [‘Uniqueness of solutions to Schrödinger equations on H-type groups’, J. Aust. Math. Soc. (3) 95 (2013), 297–314] and by Ludwig and Müller [‘Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups’, Proc. Amer. Math. Soc. 142 (2014), 2101–2118].
publishDate 2020
dc.date.none.fl_str_mv 2020
2021
2021
dc.type.none.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://hdl.handle.net/10810/52480
url http://hdl.handle.net/10810/52480
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv info:eu-repo/grantAgreement/EC/H2020/669689
info:eu-repo/grantAgreement/MINECO/MTM2014-53145-P/
https://doi.org/10.1017/S1446788720000026
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
CC BY-NC-ND
eu_rights_str_mv openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by-nc-nd/3.0/es/
CC BY-NC-ND
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Cambridge University Press
publisher.none.fl_str_mv Cambridge University Press
dc.source.none.fl_str_mv reponame:Addi. Archivo Digital para la Docencia y la Investigación
instname:Universidad del País Vasco
instname_str Universidad del País Vasco
reponame_str Addi. Archivo Digital para la Docencia y la Investigación
collection Addi. Archivo Digital para la Docencia y la Investigación
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.300724

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