Spectral Analysis for Perturbed Operators on Carnot Groups

TL;DR

This paper proves that certain fractional powers of the sublaplacian plus a potential on Carnot groups have purely absolutely continuous spectrum, using commutator methods and Hardy inequalities.

Contribution

It establishes the absence of singular spectrum for operators of the form $H_eta$ on Carnot groups under broad conditions on the potential.

Findings

Operators $H_eta$ have no singular spectrum.

The methods involve commutator techniques and Hardy inequalities.

Results apply to a class of operators on Carnot groups.

Abstract

Let $\G$ be a Carnot group of homogeneous dimension $M$ and $\Delta$ its horizontal sublaplacian. For $\alpha\in(0,M)$ we show that operators of the form $H_\alpha:=(-\Delta)^\alpha+V$ have no singular spectrum, under generous assumptions on the multiplication operator $V$. The proof is based on commutator methods and Hardy inequalities.