Uncertainty relations on nilpotent Lie groups

TL;DR

This paper extends fundamental quantum mechanical operator relations and uncertainty inequalities to a broad class of nilpotent Lie groups, providing new mathematical insights and optimal constants in these generalized settings.

Contribution

It introduces novel uncertainty relations and inequalities for quantum operators on nilpotent Lie groups, including best constants and generalizations of classical inequalities.

Findings

Derived operator relations on nilpotent Lie groups.

Established uncertainty inequalities with optimal constants.

Extended classical inequalities to new geometric settings.

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 $\mathbb R^{n}$, and of the Heisenberg group.