Resolvent Estimates Near the Boundary of the Range of the Symbol

  • Johannes Sjöstrand
Part of the Pseudo-Differential Operators book series (PDO, volume 14)


The purpose of this chapter is to give quite explicit bounds on the resolvent near the boundary of Σ(p) (or more generally, near certain “generic boundary-like” points.) The result is due (up to a small generalization) to Montrieux (Estimation de résolvante et construction de quasimode pres du bord du pseudospectre, 2013) and improves earlier results by Martinet (Sur les propriétés spectrales d’opérateurs nonautoadjoints provenant de la mécanique des fluides, 2009) about upper and lower bounds for the norm of the resolvent of the complex Airy operator, which has empty spectrum (Almog, SIAM J Math Anal 40:824–850, 2008). There are more results about upper bounds, and some of them will be recalled in Chap.  10 when dealing with such bounds in arbitrary dimension.


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Authors and Affiliations

  • Johannes Sjöstrand
    • 1
  1. 1.Université de Bourgogne Franche-ComtéDijonFrance

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