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Analysis of periodic Schrodinger operators: regularity and approximation of eigenfunctions.
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 Coulombtype 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...
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2008

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Online Access:  https://hdl.handle.net/2134/17165 
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author  Eugenie Hunsicker Victor Nistor Jorge O. Sofo 
author_facet  Eugenie Hunsicker Victor Nistor Jorge O. Sofo 
author_sort  Eugenie Hunsicker (1247667) 
collection  Figshare 
description  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 Coulombtype 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. 
format  Default Article 
id  rrarticle9385784 
institution  Loughborough University 
publishDate  2008 
record_format  Figshare 
spelling  rrarticle938578420080101T00:00:00Z Analysis of periodic Schrodinger operators: regularity and approximation of eigenfunctions. Eugenie Hunsicker (1247667) Victor Nistor (7161377) Jorge O. Sofo (1801576) Other mathematical sciences not elsewhere classified untagged Mathematical Sciences not elsewhere classified 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 Coulombtype 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. 20080101T00:00:00Z Text Journal contribution 2134/17165 https://figshare.com/articles/journal_contribution/Analysis_of_periodic_Schrodinger_operators_regularity_and_approximation_of_eigenfunctions_/9385784 CC BYNCND 4.0 
spellingShingle  Other mathematical sciences not elsewhere classified untagged Mathematical Sciences not elsewhere classified Eugenie Hunsicker Victor Nistor Jorge O. Sofo Analysis of periodic Schrodinger operators: regularity and approximation of eigenfunctions. 
title  Analysis of periodic Schrodinger operators: regularity and approximation of eigenfunctions. 
title_full  Analysis of periodic Schrodinger operators: regularity and approximation of eigenfunctions. 
title_fullStr  Analysis of periodic Schrodinger operators: regularity and approximation of eigenfunctions. 
title_full_unstemmed  Analysis of periodic Schrodinger operators: regularity and approximation of eigenfunctions. 
title_short  Analysis of periodic Schrodinger operators: regularity and approximation of eigenfunctions. 
title_sort  analysis of periodic schrodinger operators: regularity and approximation of eigenfunctions. 
topic  Other mathematical sciences not elsewhere classified untagged Mathematical Sciences not elsewhere classified 
url  https://hdl.handle.net/2134/17165 