Poincar\'e inequalities on Carnot Groups and spectral gap of Schr\"odinger operators

TL;DR

This paper establishes conditions for Poincaré inequalities on Carnot groups with applications to spectral gaps of Schrödinger operators, extending previous results and covering many known groups.

Contribution

It provides a new sufficient condition for Poincaré inequalities on Carnot groups and demonstrates its implications for spectral gaps of Schrödinger operators.

Findings

The condition applies to many Carnot groups including classical examples.

It guarantees the existence of a spectral gap for associated Schrödinger operators.

The results extend previous work by CFZ21.

Abstract

In this work we give a sufficient condition under which the global Poincar\'{e} inequality on Carnot groups holds true for a large family of probability measures absolutely continuous with respect to the Lebesgue measure. The density of such probability measure is given in terms of homogeneous quasi-norm on the group. We provide examples to which our condition applies including the most known families of Carnot groups. This, in particular, allows to extend the results in the previous work [CFZ21]. A consequence of our result is that the associated Schr\"{o}dinger operators have a spectral gap.