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Bounds on the number of eigenvalues of the Schrödinger operator

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Abstract

Inequalities on eigenvalues of the Schrödinger operator are re-examined in the case of spherically symmetric potentials. In particular, we obtain:

  1. i)

    A connection between the moments of order (n − 1)/2 of the eigenvalues of a one-dimensional problem and the total number of bound statesN n, inn space dimensions;

  2. ii)

    optimal bounds on the total number of bound states below a given energy in one dimension;

  3. iii)

    alower bound onN 2;

  4. iv)

    a self-contained proof of the inequality for α ≧ 0,n ≧ 3, leading to the optimalC 04,C 3;

  5. v)

    solutions of non-linear variation equations which lead, forn ≧ 7, to counter examples to the conjecture thatC 0n is given either by the one-bound state case or by the classic limit; at the same time a conjecture on the nodal structure of the wave functions is disproved.

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Authors and Affiliations

  1. CERN, CH-1211, Geneva 23, Switzerland

    V. Glaser, H. Grosse & A. Martin

Authors
  1. V. Glaser
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  2. H. Grosse
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  3. A. Martin
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Communicated by R. Stora

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Glaser, V., Grosse, H. & Martin, A. Bounds on the number of eigenvalues of the Schrödinger operator. Commun.Math. Phys. 59, 197–212 (1978). https://doi.org/10.1007/BF01614249

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  • DOI: https://doi.org/10.1007/BF01614249

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