Analysis of periodic Schrödinger operators: Regularity and approximation of eigenfunctions

Let V be a real valued potential that is smooth everywhere on ℝ³, except at a periodic, discrete set S of points, where it has singularities of the Coulomb-type Z/r. We assume that the potential V is periodic with period lattice L. We study the spectrum of the Schrödinger operator H=-Δ+V acting on the space of Bloch waves with arbitrary, but fixed, wavevector k. Let := ℝ³ / ℒ. Let u be an eigenfunction of H with eigenvalue λ and let ∈ > 0 be arbitrarily small. We show that the classical regularity of the eigenfunction u is u ∈ H^5/2-∈ () in the usual Sobolev spaces, and u ∈ Κ _3/2-∈^m (\S) in the weighted Sobolev spaces. The regularity index m can be as large as desired, which is crucial for numerical methods. For any choice of the Bloch wavevector k, we also show that H has compact resolvent and hence a complete eigenfunction expansion. The case of the hydrogen atom suggests that our regularity results are optimal. We present two applications to the numerical approximation of eigenvalues: using wave functions and using piecewise polynomials.