## 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

### MSC:

 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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### References:

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