Spectral shift function and resonances for non-semi-bounded and Stark Hamiltonians

Abstract

AbstractWe generalize for non-semi-bounded Schrödinger type operators the result of [Bruneau, Petkov, Duke Math. J. 116 (2003) 389–430] proving a representation of the derivative of the spectral shift function ξ(λ,h) related to the semiclassical resonances. For Stark Hamiltonians P2(h)=−h2Δ+βx1+V(x), β>0, we obtain the same result as well as a local trace formula. We establish an upper bound O(h−n) for the number of the resonances in a compact domain Ω⊂C− and we obtain a Weyl-type asymptotics of ξ(λ,h) for V∈C∞(Rn) with suppx1V⊂[R,+∞[. Finally, we establish the existence of resonances in every h-independent complex neighborhood of E0 if E0 is an analytic singularity of a suitable measure related to V