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Analysis of periodic Schrödinger operators: Regularity and approximation of eigenfunctions

J. Math. Phys. 49, 083501 (2008); doi:10.1063/1.2957940

Published 4 August 2008

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Eugenie Hunsicker,1 Victor Nistor,2 and Jorge O. Sofo3
1Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom
2Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802, USA
3Department of Physics, Pennsylvania State University, University Park, Pennsylvania 16802, USA
Let V be a real valued potential that is smooth everywhere on [openface R]3, except at a periodic, discrete set [script S] of points, where it has singularities of the Coulomb-type Z/r. We assume that the potential V is periodic with period lattice [script L]. We study the spectrum of the Schrödinger operator H=−Delta+V acting on the space of Bloch waves with arbitrary, but fixed, wavevector k. Let [openface T]:=[openface R]3/[script L]. Let u be an eigenfunction of H with eigenvalue lambda and let epsilon>0 be arbitrarily small. We show that the classical regularity of the eigenfunction u is u[is-an-element-of]H5/2−epsilon([openface T]) in the usual Sobolev spaces, and u[is-an-element-of][script K]<sub>3/2 - epsilon</sub><sup>m</sup>([openface T]\[script 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. ©2008 American Institute of Physics
History: Received 30 January 2008; accepted 24 June 2008; published 4 August 2008
Permalink: http://link.aip.org/link/?JMAPAQ/49/083501/1
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KEYWORDS and PACS

Keywords
PACS
  • 03.65.Ge
    Solutions of wave equations: bound states in quantum mechanics
  • 02.10.Ud
    Linear algebra
  • 03.65.Db
    Functional analytical methods in quantum mechanics
  • 03.65.Fd
    Algebraic methods in quantum mechanics
  • 02.60.-x
    Numerical approximation and analysis
  • 02.10.De
    Algebraic structures and number theory
  • YEAR: 2008

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ISSN:
0022-2488 (print)   1089-7658 (online)
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