Non self-adjoint perturbations of the Heisenberg sublaplacian

TL;DR

This paper establishes uniform resolvent estimates for the Heisenberg sublaplacian with complex perturbations, using multiplier methods and Hardy inequalities, ensuring spectral stability under certain conditions.

Contribution

It introduces new resolvent estimates for non self-adjoint perturbations of the Heisenberg sublaplacian, with explicit constants and applications to spectral analysis.

Findings

Uniform resolvent estimates in weighted L^2 spaces

Explicit constants depending only on the dimension d

Conditions ensuring the absence of eigenvalues for perturbed operators

Abstract

We prove uniform resolvent estimates in weighted $L^2$-spaces for the sublaplacian $\mathcal{L}$ on the Heisenberg group $\mathbb{H}^d$. The proof are based on multiplier methods, and strongly rely on the use of horizontal multipliers and the associated Hardy inequalities. The constants of our inequalities are explicit and depend only on the dimension $d$. As applications of this method, we obtain some suitable smallness and repulsivity conditions on a complex potential $V$ on $\mathbb{H}^d$ such that the spectrum of $\mathcal{L}+V$ does not contain eigenvalues.