## 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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