Semiclassical study of shape resonances in the Stark effect

Semiclassical behavior of Stark resonances is studied. The complex distortion outside a cone is introduced to study resonances in any energy region for the Stark Hamiltonians with non-globally analytic potentials. The non-trapping resolvent estimate is proved by the escape function method. The Weyl law and the resonance expansion of the propagator are proved in the shape resonance model. To prove the resonance expansion theorem, the functional pseudodifferential calculus in the Stark effect is established, which is also useful in the study of the spectral shift function.