Bound on the number of eigenvalues near the boundary of the pseudospectrum
HTML articles powered by AMS MathViewer
- by Mildred Hager PDF
- Proc. Amer. Math. Soc. 135 (2007), 3867-3873 Request permission
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.References
- E. B. Davies, Semi-classical states for non-self-adjoint Schrödinger operators, Comm. Math. Phys. 200 (1999), no. 1, 35–41. MR 1671904, DOI 10.1007/s002200050521
- E. Brian Davies, Pseudospectra of differential operators, J. Operator Theory 43 (2000), no. 2, 243–262. MR 1753408
- Nils Dencker, Johannes Sjöstrand, and Maciej Zworski, Pseudospectra of semiclassical (pseudo-) differential operators, Comm. Pure Appl. Math. 57 (2004), no. 3, 384–415. MR 2020109, DOI 10.1002/cpa.20004
- Mouez Dimassi and Johannes Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, Cambridge, 1999. MR 1735654, DOI 10.1017/CBO9780511662195
- I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142, DOI 10.1090/mmono/018
- Mildred Hager, Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints. I. Un modèle, Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), no. 2, 243–280 (French, with English and French summaries). MR 2244217, DOI 10.5802/afst.1121
- Mildred Hager, Instabilité spectrale semiclassique d’opérateurs non-autoadjoints. II, Ann. Henri Poincaré 7 (2006), no. 6, 1035–1064 (French, with English and French summaries). MR 2267057, DOI 10.1007/s00023-006-0275-7
- B. Helffer, J. Sjöstrand, Résonances en limite semi-classique, Bulletin de la Soc. Math. France (1986)
- Johannes Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J. 60 (1990), no. 1, 1–57. MR 1047116, DOI 10.1215/S0012-7094-90-06001-6
- Johannes Sjöstrand and Maciej Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc. 4 (1991), no. 4, 729–769. MR 1115789, DOI 10.1090/S0894-0347-1991-1115789-9
- J. Sjöstrand, Lectures on resonances, http://daphne.math.polytechnique.fr/˜sjoestrand/
- Johannes Sjöstrand, Resonances for bottles and trace formulae, Math. Nachr. 221 (2001), 95–149. MR 1806367, DOI 10.1002/1522-2616(200101)221:1<95::AID-MANA95>3.0.CO;2-P
- E.C. Titchmarsh, The theory of functions, Oxford University Press (1939)
- Lloyd N. Trefethen, Pseudospectra of linear operators, SIAM Rev. 39 (1997), no. 3, 383–406. MR 1469941, DOI 10.1137/S0036144595295284
- Maciej Zworski, A remark on a paper of E. B Davies: “Semi-classical states for non-self-adjoint Schrödinger operators” [Comm. Math. Phys. 200 (1999), no. 1, 35–41; MR1671904 (99m:34197)], Proc. Amer. Math. Soc. 129 (2001), no. 10, 2955–2957. MR 1840099, DOI 10.1090/S0002-9939-01-05909-3
- Maciej Zworski, Numerical linear algebra and solvability of partial differential equations, Comm. Math. Phys. 229 (2002), no. 2, 293–307. MR 1923176, DOI 10.1007/s00220-002-0683-6
Additional Information
- Mildred Hager
- Affiliation: CMLS, Ecole polytechnique, 91128 Palaiseau Cédex, France, UMR 7640
- Email: mildred.hager@math.polytechnique.fr
- Received by editor(s): July 5, 2006
- Received by editor(s) in revised form: August 18, 2006
- Published electronically: August 29, 2007
- Communicated by: Mikhail Shubin
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 3867-3873
- MSC (2000): Primary 34E10, 47G10, 47A75
- DOI: https://doi.org/10.1090/S0002-9939-07-08914-9
- MathSciNet review: 2341937