Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group

In this article, we introduce a notion of magnetic field in the Heisenberg group and study its influence on spectral properties corresponding (sub-elliptic) Laplacian. We show that uniform fields uplift bottom spectrum. For vanishing at infinity, including Aharonov–Bohm potentials, derive improvements to variety Hardy-type inequalities for sub-Laplacian. particular, establish sub-Riemannian analogue Laptev Weidl sub-criticality result Laplacians plane. Instrumental our argument is validity inequality Folland–Stein operator, prove article has an interest own.