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Analysis of periodic Schrodinger operators: regularity and approximation of eigenfunctions.

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posted on 31.03.2015, 10:37 by Eugenie Hunsicker, Victor Nistor, Jorge O. Sofo
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.

Funding

V.N. was supported in part by NSF Grant Nos. DMS 0555831, DMS 0713743, and OCI 0749202.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

J. Math. Phys.

Volume

49

Issue

8

Pages

Art No. 083501 - ?

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.

Publisher

© American Institute of Physics

Version

VoR (Version of Record)

Publication date

2008

Notes

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.

ISSN

0022-2488

Language

en

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