Analysis of Schr\"odinger operators with inverse square potentials {II}: FEM and approximation of eigenfunctions in the periodic case

Let $V$ be a {\em periodic} potential on ${\text{RR}}^3$ that is smooth everywhere except at a discrete set $\text{maS}$ of points, where it has singularities of the form $Z\mathrm{/}{\rho }^2$, with $\rho (x)=\mathrm{\mid }x-p\mathrm{\mid }$ for $x$ close to $p$ and $Z$ is continuous, $Z(p)>-1\mathrm{/}4$ for $p\in \text{maS}$. We also assume that $\rho$ and $Z$ are smooth outside $\text{maS}$ and $Z$ is smooth in polar coordinates around each singular point. Let us denote by $\mathrm{\Lambda }$ the periodicity lattice and set $\text{TT}:={\text{RR}}^3\mathrm{/}\mathrm{\Lambda }$. In the first paper of this series \cite{HLNU1}, we obtained regularity results in weighted Sobolev space for the eigenfunctions of the Schr"odinger-type operator $H=-\mathrm{\Delta }+V$ acting on $L^2(\text{TT})$, as well as for the induced $\text{vt}k$–Hamiltonians $\text{Hk}$ obtained by resticting the action of $H$ to Bloch waves. In this paper we present two related applications: one to the Finite Element approximation of the solution of $(L+\text{Hk})v=f$ and one to the numerical approximation of the eigenvalues, $\lambda$, and eigenfunctions, $u$, of $\text{Hk}$. We give optimal, higher order convergence results for approximation spaces defined piecewise polynomials. Our numerical tests are in good agreement with the theoretical results.