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

Abstract

We 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 P 2( h)=− h 2Δ+ βx 1+ 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 ∞( R n) with supp x 1 V⊂[ R,+∞[. Finally, we establish the existence of resonances in every h-independent complex neighborhood of E 0 if E 0 is an analytic singularity of a suitable measure related to V.