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Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations pp 191–209Cite as

Birkhäuser

Resolvent Estimates Near the Boundary of the Range of the Symbol

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Part of the book series: Pseudo-Differential Operators ((PDO,volume 14))

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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Notes

  1. 1.

    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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  1. Université de Bourgogne Franche-Comté, Dijon, France

    Johannes Sjöstrand

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Sjöstrand, J. (2019). Resolvent Estimates Near the Boundary of the Range of the Symbol. In: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations. Pseudo-Differential Operators, vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10819-9_10

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