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

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.