Abstract
We consider eigenvaluesE λ of the HamiltonianH λ=−Δ+V+λW,W compactly supported, in the λ→∞ limit. ForW≧0 we find monotonic convergence ofE λ 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 eigenvalueE λ 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 ofH λ 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→∞.
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Communicated by A. Jaffe
Max Kade Foundation Fellow
Partially supported by USNSF under Grant DMS-8416049
On leave of absence from Department of Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA. Partially supported by USNSF under Grant DMS-8620231 and the Case Institute of Technology, RIG
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Gesztesy, F., Gurarie, D., Holden, H. et al. Trapping and cascading of eigenvalues in the large coupling limit. Commun.Math. Phys. 118, 597–634 (1988). https://doi.org/10.1007/BF01221111
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DOI: https://doi.org/10.1007/BF01221111