Uniform resolvent estimates, smoothing effects and spectral stability for the Heisenberg sublaplacian

TL;DR

This paper establishes uniform resolvent estimates for the Heisenberg sublaplacian and its fractional powers, leading to smoothing effects and spectral stability results for related Schrödinger equations.

Contribution

It introduces a new abstract resolvent estimate using weakly conjugate operators and Hardy inequalities, improving previous results for the Heisenberg sublaplacian.

Findings

Proves Kato-type smoothing effects for Schrödinger equations on the Heisenberg group.

Shows spectral stability of the sublaplacian under certain complex potential perturbations.

Obtains uniform estimates without symmetry or derivative loss in the local case.

Abstract

We establish global bounds for solutions to stationary and time-dependent Schr\"odinger equations associated with the sublaplacian $\mathcal L$ on the Heisenberg group, as well as its pure fractional power $\mathcal L^s$ and conformally invariant fractional power $\mathcal L_s$. The main ingredient is a new abstract uniform weighted resolvent estimate which is proved by using the method of weakly conjugate operators -- a variant of Mourre's commutator method -- and Hardy's type inequalities on the Heisenberg group. As applications, we show Kato-type smoothing effects for the time-dependent Schr\"odinger equation, and spectral stability of the sublaplacian perturbed by complex-valued decaying potentials satisfying an explicit subordination condition. In the local case $s=1$, we obtain uniform estimates without any symmetry or derivative loss, which improve previous results.