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Bound on the number of eigenvalues near the boundary of the pseudospectrum
Author:
Mildred Hager
Journal:
Proc. Amer. Math. Soc. 135 (2007), 3867-3873
MSC (2000):
Primary 34E10, 47G10, 47A75
Posted:
August 29, 2007
MathSciNet review:
2341937
Full-text PDF Free Access
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Abstract: We show an estimate of the number of eigenvalues in a neighbourhood of a finite part of the boundary of the semiclassical pseudospectrum of pseudodifferential non-selfadjoint operators in terms of a corresponding volume in phase space.
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“Semi-classical states for non-self-adjoint Schrödinger
operators” [Comm. Math. Phys. 200 (1999), no. 1, 35–41;
MR1671904 (99m:34197)], Proc. Amer. Math.
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- 2.
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- 3.
- N. Dencker, J. Sjöstrand, M. Zworski, Pseudospectra of semiclassical (pseudo-) differential operators, Comm. Pure Appl. Math. 57 (2004), 384-415 MR 2020109 (2004k:35432)
- 4.
- M. Dimassi, J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit, London Math. Soc., Lecture Note Series 268, Cambridge University Press (1999) MR 1735654 (2001b:35237)
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- I.C. Gohberg, M.G. Krein, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs 18, Amer. Math. Soc. (1969) MR 0246142 (39:7447)
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- B. Helffer, J. Sjöstrand, Résonances en limite semi-classique, Bulletin de la Soc. Math. France (1986)
- 9.
- J. Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Mathematical Journal 60 (1990), 1-57 MR 1047116 (91e:35166)
- 10.
- J. Sjöstrand, M. Zworski, Complex scaling and the distribution of scattering poles, Jour. Amer. Math. Soc. 4 (1991), 729-769 MR 1115789 (92g:35166)
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- J. Sjöstrand, Lectures on resonances, http://daphne.math.polytechnique.fr/~sjoestrand/
- 12.
- J. Sjöstrand, Resonances for bottles and trace formulae, Math. Nachr. 221 (2001), 95-149. MR 1806367 (2001k:58063)
- 13.
- E.C. Titchmarsh, The theory of functions, Oxford University Press (1939)
- 14.
- L.N. Trefethen, Pseudospectra of linear operators, SIAM Rev. 39 (1997), 383-406 MR 1469941 (98i:47004)
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- M. Zworski, A remark on a paper of E.B. Davies, Proceedings of the AMS 129 (1999), 2955-2957 MR 1840099 (2002e:35015)
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Additional Information
Mildred Hager
Affiliation:
CMLS, Ecole polytechnique, 91128 Palaiseau Cédex, France, UMR 7640
Email:
mildred.hager@math.polytechnique.fr
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08914-9
PII:
S 0002-9939(07)08914-9
Keywords:
Pseudospectrum,
perturbation,
non-selfadjoint operators
Received by editor(s):
July 5, 2006
Received by editor(s) in revised form:
August 18, 2006
Posted:
August 29, 2007
Communicated by:
Mikhail Shubin
Article copyright:
© Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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