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.


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