Uncertainty relations on nilpotent Lie groups

Michael Ruzhansky, Durvudkhan Suragan

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


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
Issue number2201
Publication statusPublished - May 1 2017


  • Homogeneous Lie group
  • Nilpotent Lie group
  • Uncertainty principle

ASJC Scopus subject areas

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

Fingerprint Dive into the research topics of 'Uncertainty relations on nilpotent Lie groups'. Together they form a unique fingerprint.

Cite this