Uncertainty relations on nilpotent Lie groups

Michael Ruzhansky, Durvudkhan Suragan

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. Namely, we show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy's inequality, Heisenberg-Kennard type and Heisenberg-Pauli-Weyl type uncertainty inequalities, as well as Caffarelli-Kohn-Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic Rn, and of the Heisenberg group. The proof demonstrates that the method of establishing equalities in sharper versions of such inequalities works well in both isotropic and anisotropic settings.

Original languageEnglish
Article number20170082
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume473
Issue number2201
DOIs
Publication statusPublished - May 1 2017

Fingerprint

Lie groups
Uncertainty Relation
Nilpotent Lie Group
Quantum theory
Homogeneous Groups
Momentum
Caffarelli-Kohn-Nirenberg Inequalities
Potential Operators
Hardy Inequality
Coulomb Potential
Best Constants
operators
Heisenberg Group
Operator
Quantum Mechanics
Euler
Equality
Analogue
Uncertainty
Coulomb potential

Keywords

  • Homogeneous Lie group
  • Nilpotent Lie group
  • Uncertainty principle

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)

Cite this

Uncertainty relations on nilpotent Lie groups. / Ruzhansky, Michael; Suragan, Durvudkhan.

In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 473, No. 2201, 20170082, 01.05.2017.

Research output: Contribution to journalArticle

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