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

7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol