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Uncertainty relations on nilpotent Lie groups
Michael Ruzhansky1, Durvudkhan Suragan2,1
1Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK.
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 [Formula: see text], 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.
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