Abstract
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.
Highlights
We study the semiclassical behavior of the resonances for the Stark Hamiltonian: P ( ) = − 2∆ + βx1 + V (x), where V (x) ∈ C∞(Rn; R) is a non-globally analytic potential and β > 0
To prove the resonance expansion theorem, we study the pseudodifferential property of ψ(P )
In [19] the resolvent estimate is obtained by the abstract method based on the maximum principle technique
Summary
We study the semiclassical behavior of the resonances for the Stark Hamiltonian:. The following bound implies the bound for the analytically continued cutoff resolvent χR+(z, h)χ for Imz > −M log −1, where χ ∈ L∞ cone(Rn), since χR+(z, h)χ = χ(z − Pθ( ))−1χ if Pθ is constructed by the deformation outside suppχ. To prove the resonance expansion theorem, we study the pseudodifferential property of ψ(P ).
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