Generalized uncertainty inequalities

TL;DR

This paper develops generalized uncertainty inequalities for pairs of positive-self adjoint operators, extending classical results to various geometric and algebraic structures like manifolds, symmetric spaces, graphs, and Lie groups.

Contribution

It introduces a new framework for uncertainty inequalities based on a balance condition of spectral projectors, applicable across diverse mathematical settings.

Findings

Derived Heisenberg-Pauli-Weyl-type inequalities for operator pairs.

Extended inequalities to Riemannian manifolds and symmetric spaces.

Applied results to graphs and Lie groups with sublaplacians.

Abstract

In this paper, Heisenberg-Pauli-Weyl-type uncertainty inequalities are obtained for a pair of positive-self adjoint operators on a Hilbert space, whose spectral projectors satisfy a ``balance condition'' involving certain operator norms. This result is then applied to obtain uncertainty inequalities on Riemannian manifolds, Riemannian symmetric spaces of non-compact type, homogeneous graphs and unimodular Lie groups with sublaplacians.