Stark ladder resonances for small electric fields

We prove the existence of resonances in the semi-classical regime of small h for Stark ladder Hamiltonians {Mathematical expression} in one-dimension. The potential v 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 strength F satisfies the bounds |v′|_∞>F>0. In general, the imaginary part of the resonances are bounded above by[Figure not available: see fulltext.], for some 0<κ≦1, where ρ_Th^-1 is the single barrier tunneling distance in the Agmon metric for v+Fx. In the regime where the distance between resonant wells is {Mathematical expression}, we prove that there is at least one resonance whose width is bounded above by ce^-α/F, for some α, 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.