Analysis of Schrödinger operators with inverse square potentials {II}: FEM and approximation of eigenfunctions in the periodic case

Abstract

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