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Explicit Green operators for quantum mechanical Hamiltonians. I. The hydrogen atom

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Abstract

We study a new approach to determine the asymptotic behaviour of quantum many-particle systems near coalescence points of particles which interact via singular Coulomb potentials. This problem is of fundamental interest in electronic structure theory in order to establish accurate and efficient models for numerical simulations. Within our approach, coalescence points of particles are treated as embedded geometric singularities in the configuration space of electrons. Based on a general singular pseudo-differential calculus, we provide a recursive scheme for the calculation of the parametrix and corresponding Green operator of a nonrelativistic Hamiltonian. In our singular calculus, the Green operator encodes all the asymptotic information of the eigenfunctions. Explicit calculations and an asymptotic representation for the Green operator of the hydrogen atom and isoelectronic ions are presented.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Heinz-Jürgen Flad & Reinhold Schneider

  2. Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, 26111, Oldenburg, Germany

    Gohar Harutyunyan

  3. Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469, Potsdam, Germany

    Bert-Wolfgang Schulze

Authors
  1. Heinz-Jürgen Flad
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  2. Gohar Harutyunyan
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  3. Reinhold Schneider
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  4. Bert-Wolfgang Schulze
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Correspondence to Heinz-Jürgen Flad.

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Flad, HJ., Harutyunyan, G., Schneider, R. et al. Explicit Green operators for quantum mechanical Hamiltonians. I. The hydrogen atom. manuscripta math. 135, 497–519 (2011). https://doi.org/10.1007/s00229-011-0429-x

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  • DOI: https://doi.org/10.1007/s00229-011-0429-x

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