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Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities

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The Stability of Matter: From Atoms to Stars

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Abstract

Estimates for the number of bound states and their energies, ej ≤ 0, are of obvious importance for the investigation of quantum mechanical Hamiltonians. If the latter are of the single particle form H = -Δ + V(x) in Rn , we shall use available methods to derive the bounds

$$ {\sum\limits_j {\left| {{e_j}} \right|} ^y} \leqslant {L_{y,n}}\int {{d^n}x{{\left| {V(x)} \right|}_ - }^{y + n/2}} ,y > \max \left( {0.1 - n/2} \right). $$
((1.1))

Work supported by U. S. National Science Foundation Grant MPS 71–03375-A03.

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Author information

Authors and Affiliations

  1. Departments of Mathematics and Physics, Princeton University, Princeton, New Jersey, USA

    Elliott H. Lieb

  2. Institut für Theoretische Physik, Universität Wien, Austria

    Walter E. Thirring

Authors
  1. Elliott H. Lieb
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  2. Walter E. Thirring
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Editors and Affiliations

  1. Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, A-1090, Wien, Austria

    Walter Thirring

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Lieb, E.H., Thirring, W.E. (2001). Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities. In: Thirring, W. (eds) The Stability of Matter: From Atoms to Stars. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04360-8_16

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  • DOI: https://doi.org/10.1007/978-3-662-04360-8_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-04362-2

  • Online ISBN: 978-3-662-04360-8

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