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2019 | OriginalPaper | Buchkapitel

10. Resolvent Estimates Near the Boundary of the Range of the Symbol

verfasst von : Johannes Sjöstrand

Erschienen in: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations

Verlag: Springer International Publishing

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Abstract

In this chapter, which closely follows, we study bounds on the resolvent of a non-self-adjoint h-pseudodifferential operator P with (semi-classical) principal symbol p when h → 0, when the spectral parameter is in a neighborhood of certain points on the boundary of the range of p. In Chap. 6 we have already described a very precise result of W. Bordeaux Montrieux in dimension 1. Here we consider a more general situation; the dimension can be arbitrary and we allow for more degenerate behaviour. The results will not be quite as precise as in the one-dimensional case.

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Vorheriges Kapitel Quasi-Modes in Higher Dimension

Nächstes Kapitel From Resolvent Estimates to Semigroup Bounds

Roughly, we say that two distributions u ₁ and u ₂ of temperate growth, also with respect to h, are equal microlocally near a point in phase space, if there exists an h-pseudodifferential operator χ (used very much as a cutoff function in ordinary distribution theory) with symbol equal to 1 near that point, such that χ(u ₁ − u ₂) is \({\mathcal {O}}(h^\infty ) \) in L ². Roughly, the similar notion for operators is obtained by using cutoffs to the right and to the left. We are here at the level of ideas only and avoid formal definitions.

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Titel: Resolvent Estimates Near the Boundary of the Range of the Symbol
verfasst von: Johannes Sjöstrand
Verlag: Springer International Publishing
Buch: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations
Print ISBN: 978-3-030-10818-2

Electronic ISBN: 978-3-030-10819-9

Copyright-Jahr: 2019
DOI: https://doi.org/10.1007/978-3-030-10819-9_10

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