Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations pp 93-134 | Cite as

# Resolvent Estimates Near the Boundary of the Range of the Symbol

## Abstract

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.

## References

- 7.Y. Almog, The stability of the normal state of superconductors in the presence of electric currents. SIAM J. Math. Anal.
**40**(2), 824–850 (2008)MathSciNetCrossRefGoogle Scholar - 18.W. Bordeaux Montrieux, Estimation de résolvante et construction de quasimode près du bord du pseudospectre (2013). http://arxiv.org/abs/1301.3102
- 41.M. Dimassi, J. Sjöstrand,
*Spectral Asymptotics in the Semi-Classical Limit*. London Mathematical Society Lecture Note Series, vol. 268 (Cambridge University Press, Cambridge, 1999)Google Scholar - 42.E.M. Dyn’kin, An operator calculus based upon the Cauchy-Green formula. Zapiski Nauchn. semin. LOMI
**30**, 33–40 (1972), J. Soviet Math.**4**(4), 329–334 (1975)CrossRefGoogle Scholar - 51.A. Grigis, J. Sjöstrand,
*Microlocal Analysis for Differential Operators.*London Mathematical Society Lecture Notes Series, vol. 196 (Cambridge University Press, Cambridge, 1994)Google Scholar - 55.M. Hager, Instabilité spectrale semiclassique d’opérateurs non-autoadjoints. II. Ann. Henri Poincaré
**7**(6), 1035–1064 (2006)MathSciNetCrossRefGoogle Scholar - 81.L. Hörmander, Fourier integral operators, lectures at the Nordic summer school of mathematics, unpublished notes, 1969. Available at http://portal.research.lu.se/portal/files/50761067/Hormander_Tjorn69.pdf
- 98.J. Martinet, Sur les propriétés spectrales d’opérateurs nonautoadjoints provenant de la mécanique des fluides, Thèse de doctorat, Université de Paris Sud, 2009Google Scholar
- 102.A. Melin, J. Sjöstrand,
*Fourier Integral Operators with Complex-Valued Phase Functions.*Fourier Integral Operators and Partial Differential Equations (Colloq. Internat., Univ. Nice, Nice, 1974), pp. 120–223. Lecture Notes in Mathematics, vol. 459 (Springer, Berlin, 1975)Google Scholar - 103.A. Melin, J. Sjöstrand, Fourier integral operators with complex phase functions and parametrix for an interior boundary value problem. Commun. Partial Differ. Equ.
**1**(4), 313–400 (1976)MathSciNetCrossRefGoogle Scholar - 108.L. Nirenberg,
*A proof of the Malgrange preparation theorem*. In*Proceedings of Liverpool Singularities-Symposium, I (1969/70)*, pp. 97–105. Lecture Notes in Mathematics, vol. 192 (Springer, Berlin, 1971)Google Scholar