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Communications in Mathematical Physics

, Volume 118, Issue 4, pp 597–634 | Cite as

Trapping and cascading of eigenvalues in the large coupling limit

  • F. Gesztesy
  • D. Gurarie
  • H. Holden
  • M. Klaus
  • L. Sadun
  • B. Simon
  • P. Vogl
Article

Abstract

We consider eigenvaluesEλ of the HamiltonianHλ=−Δ+VW,W compactly supported, in the λ→∞ limit. ForW≧0 we find monotonic convergence ofEλ to the eigenvalues of a limiting operatorH (associated with an exterior Dirichlet problem), and we estimate the rate of convergence for 1-dimensional systems. In 1-dimensional systems withW≦0, or withW changing sign, we do not find convergence. Instead, we find a cascade phenomenon, in which, as λ→∞, each eigenvalueEλ stays near a Dirichlet eigenvalue for a long interval (of lengthO(\(\sqrt \lambda \))) of the scaling range, quickly drops to the next lower Dirichlet eigenvalue, stays there for a long interval, drops again, and so on. As a result, for most large values of λ the discrete spectrum ofHλ is close to that ofE, but when λ reaches a transition region, the entire spectrum quickly shifts down by one. We also explore the behavior of several explicit models, as λ→∞.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Transition Region 
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 1988

Authors and Affiliations

  • F. Gesztesy
    • 1
    • 4
  • D. Gurarie
    • 1
  • H. Holden
    • 2
  • M. Klaus
    • 3
  • L. Sadun
    • 1
  • B. Simon
    • 1
  • P. Vogl
    • 4
  1. 1.Division of Physics, Mathematics, and AstronomyCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Institute of MathematicsUniversity of TrondheimTrondheim-NTHNorway
  3. 3.Department of MathematicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  4. 4.Institute for Theoretical PhysicsUniversity of GrazGrazAustria

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