Pseudo-spectrum for a class of semi-classical operators

Karel Pravda-Starov

Bulletin de la Société Mathématique de France (2008)

  • Volume: 136, Issue: 3, page 329-372
  • ISSN: 0037-9484

Abstract

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We study in this paper a notion of pseudo-spectrum in the semi-classical setting called injectivity pseudo-spectrum. The injectivity pseudo-spectrum is a subset of points in the complex plane where there exist some quasi-modes with a precise rate of decay. For that reason, these values can be considered as some ‘almost eigenvalues’ in the semi-classical limit. We are interested here in studying the absence of injectivity pseudo-spectrum, which is characterized by a global a priori estimate. We prove in this paper a sharp global subelliptic a priori estimate for a class of pseudo-differential operators with respect to the regularity of their symbols. Our main result extends the a priori estimate of Dencker, Sjöstrand and Zworski for a class of pseudo-differential operators with symbols of limited smoothness violating the condition ( P ) .

How to cite

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Pravda-Starov, Karel. "Pseudo-spectrum for a class of semi-classical operators." Bulletin de la Société Mathématique de France 136.3 (2008): 329-372. <http://eudml.org/doc/272389>.

@article{Pravda2008,
abstract = {We study in this paper a notion of pseudo-spectrum in the semi-classical setting called injectivity pseudo-spectrum. The injectivity pseudo-spectrum is a subset of points in the complex plane where there exist some quasi-modes with a precise rate of decay. For that reason, these values can be considered as some ‘almost eigenvalues’ in the semi-classical limit. We are interested here in studying the absence of injectivity pseudo-spectrum, which is characterized by a global a priori estimate. We prove in this paper a sharp global subelliptic a priori estimate for a class of pseudo-differential operators with respect to the regularity of their symbols. Our main result extends the a priori estimate of Dencker, Sjöstrand and Zworski for a class of pseudo-differential operators with symbols of limited smoothness violating the condition $(P)$.},
author = {Pravda-Starov, Karel},
journal = {Bulletin de la Société Mathématique de France},
keywords = {subelliptic estimate; symbols with limited smoothness; condition $(\overline\{\Psi \})$; Wick quantization},
language = {eng},
number = {3},
pages = {329-372},
publisher = {Société mathématique de France},
title = {Pseudo-spectrum for a class of semi-classical operators},
url = {http://eudml.org/doc/272389},
volume = {136},
year = {2008},
}

TY - JOUR
AU - Pravda-Starov, Karel
TI - Pseudo-spectrum for a class of semi-classical operators
JO - Bulletin de la Société Mathématique de France
PY - 2008
PB - Société mathématique de France
VL - 136
IS - 3
SP - 329
EP - 372
AB - We study in this paper a notion of pseudo-spectrum in the semi-classical setting called injectivity pseudo-spectrum. The injectivity pseudo-spectrum is a subset of points in the complex plane where there exist some quasi-modes with a precise rate of decay. For that reason, these values can be considered as some ‘almost eigenvalues’ in the semi-classical limit. We are interested here in studying the absence of injectivity pseudo-spectrum, which is characterized by a global a priori estimate. We prove in this paper a sharp global subelliptic a priori estimate for a class of pseudo-differential operators with respect to the regularity of their symbols. Our main result extends the a priori estimate of Dencker, Sjöstrand and Zworski for a class of pseudo-differential operators with symbols of limited smoothness violating the condition $(P)$.
LA - eng
KW - subelliptic estimate; symbols with limited smoothness; condition $(\overline{\Psi })$; Wick quantization
UR - http://eudml.org/doc/272389
ER -

References

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  1. [1] J.-M. Bony – « Sur l’inégalité de Fefferman-Phong », in Seminaire: Équations aux Dérivées Partielles, 1998–1999, Sémin. Équ. Dériv. Partielles, exposé no III, École Polytech., 1999, p. 16. Zbl1086.35529MR1721321
  2. [2] A. Boulkhemair – « L 2 estimates for Weyl quantization », J. Funct. Anal.165 (1999), p. 173–204. Zbl0934.35217MR1696697
  3. [3] E. B. Davies – « Pseudo-spectra, the harmonic oscillator and complex resonances », R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.455 (1999), p. 585–599. Zbl0931.70016MR1700903
  4. [4] —, « Semi-classical states for non-self-adjoint Schrödinger operators », Comm. Math. Phys.200 (1999), p. 35–41. Zbl0921.47060MR1671904
  5. [5] N. Dencker, J. Sjöstrand & M. Zworski – « Pseudospectra of semiclassical (pseudo-) differential operators », Comm. Pure Appl. Math.57 (2004), p. 384–415. Zbl1054.35035MR2020109
  6. [6] L. Hörmander – « A class of hypoelliptic pseudodifferential operators with double characteristics », Math. Ann.217 (1975), p. 165–188. Zbl0306.35032MR377603
  7. [7] —, The analysis of linear partial differential operators, Springer, 1983-1985. Zbl1115.35005
  8. [8] N. Lerner – « Energy methods via coherent states and advanced pseudo-differential calculus », in Multidimensional complex analysis and partial differential equations (São Carlos, 1995), Contemp. Math., vol. 205, Amer. Math. Soc., 1997, p. 177–201. Zbl0885.35152MR1447224
  9. [9] —, « The Wick calculus of pseudo-differential operators and some of its applications », Cubo Mat. Educ.5 (2003), p. 213–236. Zbl05508173MR1957713
  10. [10] K. Pravda-Starov – « A complete study of the pseudo-spectrum for the rotated harmonic oscillator », J. London Math. Soc. (2) 73 (2006), p. 745–761. Zbl1106.34060MR2241978
  11. [11] L. N. Trefethen – « Pseudospectra of linear operators », SIAM Rev.39 (1997), p. 383–406. Zbl0896.15006MR1469941
  12. [12] M. Zworski – « Numerical linear algebra and solvability of partial differential equations », Comm. Math. Phys.229 (2002), p. 293–307. Zbl1021.35077MR1923176

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