Low frequency asymptotics and local energy decay for the Schrödinger equation

We prove low frequency resolvent estimates and local energy decay for the Schrödinger equation in an asymptotically Euclidean setting. More precisely, we go beyond the optimal estimates by comparing the resolvent of the perturbed Schrödinger operator with the resolvent of the free Laplacian. This gi...

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Bibliographic Details
Published inJournal of the European Mathematical Society : JEMS Vol. 27; no. 8; pp. 3149 - 3190
Main Author Royer, Julien
Format Journal Article
LanguageEnglish
Published European Mathematical Society 01.01.2024
Subjects
Analysis of PDEs
Mathematical Physics
Mathematics
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ISSN1435-9855
1435-9863
1435-9863
DOI10.4171/jems/1429

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Summary:We prove low frequency resolvent estimates and local energy decay for the Schrödinger equation in an asymptotically Euclidean setting. More precisely, we go beyond the optimal estimates by comparing the resolvent of the perturbed Schrödinger operator with the resolvent of the free Laplacian. This gives the leading term for the development of this resolvent when the spectral parameter is close to 0. For this, we show in particular how we can apply the usual commutators method for generalized resolvents and simultaneously for different operators. Then we deduce similar results for the large time asymptotics of the corresponding evolution problem. Even if we are interested in this paper in the standard Schrödinger equation, we provide a method which can be applied to more general non-selfadjoint (dissipative) operators.
ISSN:1435-9855
1435-9863
1435-9863
DOI:10.4171/jems/1429