Abstract
We prove the existence of resonances in the semi-classical regime of smallh for Stark ladder Hamiltonians\(H(h,F) \equiv - h^2 \frac{{d^2 }}{{dx^2 }} + v + Fx\) in one-dimension. The potentialv is a real periodic function with period τ which is the restriction to ℝ of a function analytic in a strip about ℝ. The electric field strengthF satisfies the bounds |v′|∞>F>0. In general, the imaginary part of the resonances are bounded above by
, for some 0<κ≦1, whereρ T h -1 is the single barrier tunneling distance in the Agmon metric forv+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 byce −α/F, for some α,c>0 independent ofh andF forh sufficiently small. This is an extension of the Oppenheimer formula for the Stark effect to the case of periodic potentials.
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Communicated by B. Simon
Partially supported by NSF Grant DMS-8911242
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Combes, JM., Hislop, P.D. Stark ladder resonances for small electric fields. Commun.Math. Phys. 140, 291–320 (1991). https://doi.org/10.1007/BF02099501
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DOI: https://doi.org/10.1007/BF02099501