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

TL;DR

This paper develops and analyzes finite element methods for approximating eigenfunctions and eigenvalues of Schrödinger operators with inverse square singularities in a periodic setting, achieving optimal convergence rates.

Contribution

It introduces higher order finite element approximation techniques for Schrödinger operators with inverse square singularities and provides rigorous convergence analysis and numerical validation.

Findings

Optimal convergence rates for finite element approximations.

Numerical results confirm theoretical predictions.

Effective handling of inverse square singularities in periodic Schrödinger operators.

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 $Z/\rho^2$, with $\rho(x) = |x - p|$ for $x$ close to $p$ and $Z$ is continuous, $Z(p) > -1/4$ for $p \in \maS$. We also assume that $\rho$ and $Z$ are smooth outside $\maS$ and $Z$ is smooth in polar coordinates around each singular point. Let us denote by $\Lambda$ the periodicity lattice and set $\TT := \RR^3/ \Lambda$. 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 $H = -\Delta + V$ 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, $\lambda$, 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.