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Communications in Mathematical Physics

, Volume 59, Issue 2, pp 197–212 | Cite as

Bounds on the number of eigenvalues of the Schrödinger operator

  • V. Glaser
  • H. Grosse
  • A. Martin
Article

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 statesNn, 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 onN2;

     
  4. iv)

    a self-contained proof of the inequality for α ≧ 0,n ≧ 3, leading to the optimalC04,C3;

     
  5. v)

    solutions of non-linear variation equations which lead, forn ≧ 7, to counter examples to the conjecture thatC0n 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.

     

Keywords

Neural Network Statistical Physic Wave Function Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • V. Glaser
    • 1
  • H. Grosse
    • 1
  • A. Martin
    • 1
  1. 1.CERNGeneva 23Switzerland

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