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Existence of Stark ladder resonances. (English) Zbl 0651.47006

This paper deals with a Hamiltonian of the form: \[ H(k,t)u(x)=- u''(x)+q(x-it)u(x)+k(x-it)u(x) \] acting in \(L^ 2(R)\), where k, \(t\in R\) and \(q(x)=\sum^{N}_{n=-N}c_ ne^{inx}\) with \(c_ 0=0\) and \(c_ n\neq 0\) for at least one n. It is known that \(\tau_{ess}(H(k,t))\subset R-ikt\) and that \(\tau_{disc}(H(k,t))\subset \{z:\quad -ikt\leq Im(z)\leq 0\}.\) The present contribution shows that there exists \(k_ 0>0\) such that for each \(k>k_ 0\) there exists \(t\in R\) such that H(k,t) has an eigenvalue (and by translation an infinity of eigenvalues) not on the line R-ikt. The authors conjecture that \(k_ 0=0\).
Reviewer: C.A.Stuart

MSC:

47A55 Perturbation theory of linear operators
47E05 General theory of ordinary differential operators
81Q15 Perturbation theories for operators and differential equations in quantum theory
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