Trapping and Cascading of Eigenvalues In the Large Coupling Limit

We consider eigenvalues Eλ of the Hamiltonian Hλ= — Δ+ V+ λW, W compactly supported, in the / -> oo limit. For W ̂ 0 we find monotonic convergence of Eλ to the eigenvalues of a limiting operator H^ (associated with an exterior Dirichlet problem), and we estimate the rate of convergence for 1-dimensional systems. In 1-dimensional systems with W^09 or with W changing sign, we do not find convergence. Instead, we find a cascade phenomenon, in which, as Λ,->oo, each eigenvalue Eλ stays near a Dirichlet eigenvalue for a long interval (of length 0(^/1)) 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 of Hλ is close to that of H^, 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-+ oo.