TY - JOUR

T1 - Analysis of periodic Schrödinger operators

T2 - Regularity and approximation of eigenfunctions

AU - Hunsicker, Eugenie

AU - Nistor, Victor

AU - Sofo, Jorge O.

N1 - Funding Information:
We would like to thank Hengguang Li and Anna Mazzucato for useful discussions. V.N. and E.H. would like to thank the Max Planck Institute for Mathematics in Bonn, where part of this work was completed. We also thank the referee for carefully reading our paper. V.N. was supported in part by NSF Grant Nos. DMS 0555831, DMS 0713743, and OCI 0749202. We are grateful to Hundertmark for suggesting Refs. .

PY - 2008

Y1 - 2008

N2 - 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 ∈ H5/2-∈ () 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.

AB - 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 ∈ H5/2-∈ () 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.

UR - http://www.scopus.com/inward/record.url?scp=50849106692&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=50849106692&partnerID=8YFLogxK

U2 - 10.1063/1.2957940

DO - 10.1063/1.2957940

M3 - Article

AN - SCOPUS:50849106692

SN - 0022-2488

VL - 49

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 8

M1 - 083501

ER -