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Trapping and cascading of eigenvalues in the large coupling limit. (English) Zbl 0669.35085
Authors’ abstract: We consider eigenvalues \(E_{\lambda}\) of the Hamiltonian \(H_{\lambda}=-\Delta +V+\lambda W,\) W compactly supported, in the \(\lambda\) \(\to \infty\) limit. For \(W>0\) we find monotonic convergence of \(E_{\lambda}\) to the eigenvalues of a limiting operator \(H_{\infty}\) (associated with an exterior Dirichlet problem), and we estimate the rate of convergence for 1-dimensional systems. In 1- dimensional systems with \(W\leq 0\), or with W changing sign, we do not find convergence. Instead, we find a cascade phenomenon, in which, as \(\lambda\) \(\to \infty\), each eigenvalue \(E_{\lambda}\) stays near a Dirichlet eigenvalue for a long interval (of length O(\(\sqrt{\lambda}))\) 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 \(\lambda\) the discrete spectrum of \(H_{\lambda}\) is close to that of \(H_{\infty}\), but when \(\lambda\) reaches a transition region, the entire spectrum quickly shifts down by one. We also explore the behaviour of several explicit models, as \(\lambda\) \(\to \infty\).
Reviewer: W.D.Frans

MSC:
35P05 General topics in linear spectral theory for PDEs
35J25 Boundary value problems for second-order elliptic equations
47A55 Perturbation theory of linear operators
35Q99 Partial differential equations of mathematical physics and other areas of application
81Q15 Perturbation theories for operators and differential equations in quantum theory
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