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Sharp Regularity Results for Coulombic Many-Electron Wave Functions

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Abstract

We show that electronic wave functions ψ of atoms and molecules have a representation ψ=ϕ, where is an explicit universal factor, locally Lipschitz, and independent of the eigenvalue and the solution ψ itself, and ϕ has second derivatives which are locally in L∞. This representation turns out to be optimal as can already be demonstrated with the help of hydrogenic wave functions. The proofs of these results are, in an essential way, based on a new elliptic regularity result which is of independent interest. Some identities that can be interpreted as cusp conditions for second order derivatives of ψ are derived.

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Authors and Affiliations

  1. CNRS and Laboratoire de Mathématiques, UMR CNRS 8628, Université Paris-Sud - Bât 425, 91405, Orsay Cedex, France

    Søren Fournais

  2. Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090, Vienna, Austria

    Maria Hoffmann-Ostenhof

  3. The Erwin Schrödinger International Institute for Mathematical Physics, Boltzmanngasse 9, 1090, Vienna, Austria

    Thomas Hoffmann-Ostenhof

  4. Institut für Theoretische Chemie, Universität Wien, Währingerstraße 17, 1090, Vienna, Austria

    Thomas Hoffmann-Ostenhof

  5. Mathematisches Institut, Universität München, Theresienstraße 39, 80333, Munich, Germany

    Thomas Østergaard Sørensen

Authors
  1. Søren Fournais
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  2. Maria Hoffmann-Ostenhof
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  3. Thomas Hoffmann-Ostenhof
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  4. Thomas Østergaard Sørensen
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Correspondence to Søren Fournais.

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Communicated by B. Simon

© 2004 by the authors. This article may be reproduced in its entirety for non-commercial purposes.

On leave from Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg East, Denmark

Acknowledgement All four authors thank the organizers of the program Partial Differential Equations and Spectral Theory for invitations to the Mittag-Leffler Institute in 2002 where part of the work was done. Furthermore, parts of this work have been carried out at various institutions, whose hospitality is gratefully acknowledged: Aalborg University (SF, MHO, THO), The Erwin Schrödinger Institute (TØS), Université Paris-Sud (TØS), and the IHÉS (TØS). Financial support from the Danish Natural Science Research Council, European Science Foundation Programme Spectral Theory and Partial Differential Equations (SPECT), and EU IHP network Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems, contract no. HPRN-CT-2002-00277 is gratefully acknowledged. SF has been supported by a Marie Curie Fellowship of the European Community Programme ‘Improving the Human Research Potential and the Socio-Economic Knowledge Base’ under contract number HPMF-CT-2002-01822, and by a grant from the Carlsberg Foundation. Finally, SF and TØS wish to thank I. Herbst for useful discussions at the Mittag-Leffler Institute.

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Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T. et al. Sharp Regularity Results for Coulombic Many-Electron Wave Functions. Commun. Math. Phys. 255, 183–227 (2005). https://doi.org/10.1007/s00220-004-1257-6

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