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


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→∞.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Transition Region 
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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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