Springer Online Journal Archives 1860-2000
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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