JournalsjstVol. 3, No. 3pp. 399–422

Semiclassical estimates of the cut-off resolvent for trapping perturbations

  • Jean-François Bony

    Université Bordeaux I, Talence, France
  • Vesselin Petkov

    Université Bordeaux I, Talence, France
Semiclassical estimates of the cut-off resolvent for trapping perturbations cover
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Abstract

This paper is devoted to the study of the cut-off resolvent of a semiclassical "black box'' operator PP. We estimate the norm of φ(Pz)1φ\varphi ( P - z )^{- 1} \varphi, for any φC0(Rn)\varphi \in C^{\infty}_{0} ( \mathbb R^{n} ), by the norm of \mathds1Ca,b(Pz)1\mathds1Ca,b{\mathds{1}}_{\mathcal C_{a,b}} ( P - z )^{- 1} {\mathds{1}}_{\mathcal C_{a,b}} where Ca,b={xRn; a<x<b}\mathcal C_{a,b} = \{ x \in \mathbb R^{n} ; \ a < \vert x \vert < b \} and a1a \gg 1. For zz in the unphysical sheet with MhlnhImz0- M h \vert \ln h \vert \leq \operatorname{Im} z \leq 0, we prove that this estimate holds with a constant hImzeCImz/h\frac{h}{\vert \operatorname{Im} z \vert} e^{C \vert \operatorname{Im} z \vert / h}. We also study the resonant states uu of the operator PP and we obtain bounds for φu\Vert \varphi u \Vert by \mathds1Ca,bu\Vert \mathds{1}_{\mathcal C_{a,b}} u \Vert. These results hold without any assumption on the trapped set nor any assumption on the multiplicity of the resonances.

Cite this article

Jean-François Bony, Vesselin Petkov, Semiclassical estimates of the cut-off resolvent for trapping perturbations. J. Spectr. Theory 3 (2013), no. 3, pp. 399–422

DOI 10.4171/JST/49