Loughborough University
Leicestershire, UK
LE11 3TU
+44 (0)1509 263171
Loughborough University

Loughborough University Institutional Repository

Please use this identifier to cite or link to this item: https://dspace.lboro.ac.uk/2134/17165

Title: Analysis of periodic Schrodinger operators: regularity and approximation of eigenfunctions.
Authors: Hunsicker, Eugenie
Nistor, Victor
Sofo, Jorge O.
Issue Date: 2008
Publisher: © American Institute of Physics
Citation: HUNSICKER, E., NISTOR, V. and SOFO, J.O., 2008. Analysis of periodic Schrodinger operators: regularity and approximation of eigenfunctions. Journal of Mathematical Physics, 49 (8), 083501.
Abstract: Let V be a real valued potential that is smooth everywhere on R 3 , 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 T≔R 3 /L . Let u be an eigenfunction of H with eigenvalueλ and let ϵ>0 be arbitrarily small. We show that the classical regularity of the eigenfunction u is u∊H 5/2−ϵ (T) in the usual Sobolev spaces, and u∊K m 3/2−ϵ (T\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.
Description: Copyright 2008 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Journal of Mathematical Physics, 49 (8), 083501 and may be found at http://dx.doi.org/10.1063/1.2957940.
Sponsor: V.N. was supported in part by NSF Grant Nos. DMS 0555831, DMS 0713743, and OCI 0749202.
Version: Published
DOI: 10.1063/1.2957940
URI: https://dspace.lboro.ac.uk/2134/17165
Publisher Link: http://dx.doi.org/10.1063/1.2957940
ISSN: 0022-2488
Appears in Collections:Published Articles (Maths)

Files associated with this item:

File Description SizeFormat
1.2957940.pdfPublished version558.92 kBAdobe PDFView/Open


SFX Query

Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.