On the absence of resonances for Schrödinger operators with non- trapping potentials in the classical limit. (English) Zbl 0651.47007

This paper provides a bound for the resolvent of a dilated Schrödinger operator associated with an exterior scaling. The operator has the form \(H=-k^ 4\Delta +V\) and is considered on the exterior \((| x| >b)\) of a ball in \(R^ N\) with the Dirichlet condition on the boundary. In polar coordinates, a rescaling is defined by the transformations \(r_ t=b+e^ t(r-b)\) and \((U(t)f)(r,w)=e^{t/2}f(r_ t,w)\) where \(f\in L^ 2(| x| >b)\) and \((r,w)\in (0,\infty)\times S^{N-1}\). Given a non-trapping energy \(\epsilon\) for V, an estimate for \(| | (H(t)- z)^{-1}| |\) is established in a k-independent neighbourhood of E.
This type of question is motivated by work on the shape resonance problem and a related treatment of predissociation. The present contribution covers potentials that are not (-\(\Delta)\)-compact.
Reviewer: C.A.Stuart


47A55 Perturbation theory of linear operators
47F05 General theory of partial differential operators
81Q15 Perturbation theories for operators and differential equations in quantum theory
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