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AN UNCERTAINTY PRINCIPLE FOR SOLUTIONS OF THE SCHRÖDINGER EQUATION ON $H$-TYPE GROUPS
Published online by Cambridge University Press: 02 April 2020
Abstract
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].
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- © 2020 Australian Mathematical Publishing Association Inc.
Footnotes
Communicated by C. Meaney
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.