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 H(h, F) = -h2 [GRAPHICS] + upsilon + Fx in one-dimension. The potential-upsilon is a real periodic function with period tau which is the restriction to R of a function analytic in a strip about R. The electric field strength F satisfies the bounds parallel-to upsilon' parallel-to infinity > F > 0. In general, the imaginary part of the resonances are bounded above by ce-kappa-rho-Th-1, for some 0 < kappa less-than-or-equal-to 1, where rho-Th-1 is the single barrier tunneling distance in the Agmon metric for upsilon + Fx. In the regime where the distance between resonant wells is 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.