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.