Abstract: The talk is devoted to Stark-Wannier ladders i.e. the resonances of a one dimensional periodic operator in a constant electric field. The periodic sequences of points in the lower half of the complex plane have been conjecture to be very sensitive to the nulmber theoretical properties of the electric field. Computing the asymptotics of the reflection coefficients in the case of simple 1-periodic potential, we related the resonances to cubic exponential sums in which the frequency is computed from the electric field. In the case of rational frequency, we derive "large imarginary part" asymptotics for the resonances.

The talk is based on joint work with A. Fedotov (St Petersburg).