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Stark ladder resonances for small electric fields. (English) Zbl 0737.34060

Summary: We prove the existence of resonances in the semi-classical regime of small \(h\) for Stark ladder Hamiltonians \(H(h,F)\equiv-h^ 2{d^ 2 \over dx^ 2}+v+Fx\) in one dimension. The potential \(v\) is a real periodic function with period \(\tau\) which is the restriction to \(\mathbb{R}\) of a function analytic in a strip about \(\mathbb{R}\). The electric field strength \(F\) satisfies the bounds \(\| v'\|_ \infty>F>0\). In general, the imaginary part of the resonances are bounded above by \(ce^{- \kappa\rho_ T h^{-1}}\), for some \(0<\kappa\leq 1\), where \(\rho_ T h^{-1}\) is the single barrier tunneling distance in the Agmon metric for \(v+Fx\). In the regime where the distance between resonant wells is \({\mathcal O}(F^{-1})\), we prove that there is at least one resonance whose width is bounded above by \(ce^{-\alpha/F}\), for some \(\alpha,c>0\) independent of \(h\) and \(F\) for \(h\) sufficiently small. This is an extension of the Oppenheimer formula for the Stark effect to the case of periodic potentials.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34C25 Periodic solutions to ordinary differential equations
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