We will consider some Hardy-type inequalities for the Dirac operator
$\mathcal D$ in 3d: they are a very useful tool to construct self-adjoint
extensions for $\mathcal D + V$, where $V$ is a general Hermitian matrix
valued potential $V$, such that $|V(x)| \sim 1/|x|$, and they have been
exploited to prove the validity of min-max formulas determining the
eigenvalues of $\mathcal D + V$ in the spectral gap.
In a recent work in collaboration with Fabio Pizzichillo and Luis Vega
(Basque Center for Applied Mathematics, Bilbao) we prove a sharp
Hardy-type inequality for the Dirac operator and we exploit this to obtain
spectral properties of the Dirac operator perturbed with Hermitian matrix
valued potentials $V$ such that $\sup_x |x||V(x)| <1$: we characterise
its eigenvalues in terms of the Birman--Schwinger principle and we bound
its discrete spectrum from below, showing that the \emph{ground-state
energy} is reached if and only if $V$ verifies some {rigidity} conditions.
In the particular case of an electrostatic potential, these imply that $V$
is the Coulomb potential. We will describe these results, and finally we
will discuss some possibile generalisations to the case that $\sup_x
|x||V(x)| =1$.