Analysis of Periodic Schrödinger Operators: Regularity and Approximation of Eigenvalues

Let V be a real valued potential that is smooth everywhere on R3, except at a periodic, discrete set of points, where it has singularities of the Coulomb form Z/r. We assume that the potential V is periodic with period lattice L. We study the spectrum of the Schrödinger operator H = −∆ + V acting on the space of Bloch waves with arbitrary, but fixed, wavevector k. Let T := R3/L. Let u be an eigenvector of H with eigenvalue λ and let > 0 be arbitrarily small. We show that the classical regularity of the eigenvector u is u ∈ H5/2− (T) in the usual Sobolev spaces, and u ∈ Km 3/2− (T r 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.