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Analytic Structure of Many-Body Coulombic Wave Functions

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Abstract

We investigate the analytic structure of solutions of non-relativistic Schrödinger equations describing Coulombic many-particle systems. We prove the following: Let ψ(x) with \({{\bf x} = (x_{1},\dots, x_{N})\in \mathbb {R}^{3N}}\) denote an N-electron wavefunction of such a system with one nucleus fixed at the origin. Then in a neighbourhood of a coalescence point, for which x 1 = 0 and the other electron coordinates do not coincide, and differ from 0, ψ can be represented locally as ψ(x) = ψ (1)(x) + |x 1|ψ (2)(x) with ψ (1), ψ (2) real analytic. A similar representation holds near two-electron coalescence points. The Kustaanheimo-Stiefel transform and analytic hypoellipticity play an essential role in the proof.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 1530, DK-8000, Århus C, Denmark

    Søren Fournais

  2. Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090, Vienna, Austria

    Maria Hoffmann-Ostenhof

  3. Institut für Theoretische Chemie, Universität Wien, Währingerstrasse 17, A-1090, Vienna, Austria

    Thomas Hoffmann-Ostenhof

  4. The Erwin Schrödinger International Institute for Mathematical Physics, Boltzmanngasse 9, A-1090, Vienna, Austria

    Thomas Hoffmann-Ostenhof

  5. Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220, Aalborg East, Denmark

    Thomas Østergaard Sørensen

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  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 Thomas Østergaard Sørensen.

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Communicated by B. Simon

© 2008 by the authors. This article may be reproduced in its entirety for noncommercial purposes.

On leave from: CNRS and Laboratoire de Mathématiques d’Orsay, Univ Paris-Sud, F-91405 Orsay CEDEX, France

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Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T. et al. Analytic Structure of Many-Body Coulombic Wave Functions. Commun. Math. Phys. 289, 291–310 (2009). https://doi.org/10.1007/s00220-008-0664-5

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  • DOI: https://doi.org/10.1007/s00220-008-0664-5

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