**Abstract:** In this talk we discuss Hardy-type inequalities for
Schr\"{o}dinger operators of the form
$A_\lambda:=-\Delta -\lambda V$, $\lambda>0$, in smooth domains in
$\mathbb{R}^N$, $N\geq 1$, that are of big interest in applications to
Quantum Mechanics and related fields. The admissible range of parameters
$\lambda$ is characterized through the corresponding Hardy inequality.
In addition, due to the presence of the singularity, standard elliptic
regularity of the Dirichlet problem associated to $A_\lambda$ fails. The
loss of regularity could also be noticed at the numerical level when
looking for the convergence rates of the finite element schemes associated to
$A-\lambda$. We will present some numerical simulations in that sense.