Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics

Michael Hitrik[1]; Karel Pravda-Starov[2]

  • [1] Department of Mathematics University of California Los Angeles CA 90095-1555, USA
  • [2] Département de Mathématiques Université de Cergy-Pontoise Site de St Martin, 2 avenue Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 3, page 985-1032
  • ISSN: 0373-0956

Abstract

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For a class of non-selfadjoint h –pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we give a precise description of the spectrum of the operator in an 𝒪 ( h ) –neighborhood of the origin. Moreover, when all the singular spaces are reduced to zero, we establish accurate semiclassical resolvent estimates of subelliptic type, which depend directly on algebraic properties of the Hamilton maps associated to the quadratic approximations of the principal symbol.

How to cite

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Hitrik, Michael, and Pravda-Starov, Karel. "Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics." Annales de l’institut Fourier 63.3 (2013): 985-1032. <http://eudml.org/doc/275650>.

@article{Hitrik2013,
abstract = {For a class of non-selfadjoint $h$–pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we give a precise description of the spectrum of the operator in an $\mathcal\{O\}(h)$–neighborhood of the origin. Moreover, when all the singular spaces are reduced to zero, we establish accurate semiclassical resolvent estimates of subelliptic type, which depend directly on algebraic properties of the Hamilton maps associated to the quadratic approximations of the principal symbol.},
affiliation = {Department of Mathematics University of California Los Angeles CA 90095-1555, USA; Département de Mathématiques Université de Cergy-Pontoise Site de St Martin, 2 avenue Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France},
author = {Hitrik, Michael, Pravda-Starov, Karel},
journal = {Annales de l’institut Fourier},
keywords = {non-selfadjoint operator; eigenvalue; resolvent estimate; subelliptic estimates; double characteristics; singular space; pseudodifferential calculus; Wick calculus; FBI transform; Grushin problem; non-selfadjoint -pseudodifferential operators; eigenvalues},
language = {eng},
number = {3},
pages = {985-1032},
publisher = {Association des Annales de l’institut Fourier},
title = {Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics},
url = {http://eudml.org/doc/275650},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Hitrik, Michael
AU - Pravda-Starov, Karel
TI - Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 3
SP - 985
EP - 1032
AB - For a class of non-selfadjoint $h$–pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we give a precise description of the spectrum of the operator in an $\mathcal{O}(h)$–neighborhood of the origin. Moreover, when all the singular spaces are reduced to zero, we establish accurate semiclassical resolvent estimates of subelliptic type, which depend directly on algebraic properties of the Hamilton maps associated to the quadratic approximations of the principal symbol.
LA - eng
KW - non-selfadjoint operator; eigenvalue; resolvent estimate; subelliptic estimates; double characteristics; singular space; pseudodifferential calculus; Wick calculus; FBI transform; Grushin problem; non-selfadjoint -pseudodifferential operators; eigenvalues
UR - http://eudml.org/doc/275650
ER -

References

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  1. W. Bordeaux Montrieux, Loi de Weyl presque sûre et résolvante pour des opérateurs différentiels non-autoadjoints, (2008) 
  2. N. Dencker, J. Sjöstrand, M. Zworski, Pseudo-spectra of semiclassical (pseudo)differential operators, Comm. Pure Appl. Math. 57 (2004), 384-415 Zbl1054.35035MR2020109
  3. M. Dimassi, J. Sjöstrand, Spectral asymptotics in the semi-classical limit, (1999), Cambridge University Press Zbl0926.35002MR1735654
  4. M. Hager, J. Sjöstrand, Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators, Math. Annalen 342 (2008), 177-243 Zbl1151.35063MR2415321
  5. B. Helffer, F. Nier, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten laplacians, 1862 (2005), Springer Verlag Zbl1072.35006MR2130405
  6. B. Helffer, J. Sjöstrand, Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum, Mem. Soc. Math. France (N.S.) 39 (1989), 1-124 Zbl0725.34099MR1041490
  7. M. Hitrik, K. Pravda-Starov, Spectra and semigroup smoothing for non-elliptic quadratic operators, Math. Ann. 334 (2009), 801-846 Zbl1171.47038MR2507625
  8. M. Hitrik, K. Pravda-Starov, Semiclassical hypoelliptic estimates for non-selfadjoint operators with double characteristics, Comm. P.D.E. 35 (2010), 988-1028 Zbl1247.35015MR2753626
  9. F. Hérau, M. Hitrik, J. Sjöstrand, Tunnel effect for Kramers-Fokker-Planck type operators, Ann. Henri Poincaré 9 (2008), 209-274 Zbl1141.82011MR2399189
  10. F. Hérau, F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high degree potential, Arch. Ration. Mech. Anal. 171 (2004), 151-218 Zbl1139.82323MR2034753
  11. F. Hérau, J. Sjöstrand, C. Stolk, Semiclassical analysis for the Kramers-Fokker-Planck equation, Comm. PDE 30 (2005), 689-760 Zbl1083.35149MR2153513
  12. L. Hörmander, The analysis of linear partial differential operators, I–IV (1985), Springer-Verlag Zbl0601.35001
  13. L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z. 219 (1995), 413-449 Zbl0829.35150MR1339714
  14. N. Lerner, The Wick calculus of pseudodifferential operators and some of its applications, Cubo Mat. Educ. 5 (2003), 213-236 Zbl05508173MR1957713
  15. N. Lerner, Some Facts About the Wick Calculus, Pseudo-differential operators. Quantization and signals 1949 (2008), 135-174, Springer-Verlag, Berlin Zbl1180.35596MR2477145
  16. N. Lerner, Metrics on the phase space and non-selfadjoint pseudo-differential operators, 3 (2010), Birkhäuser Verlag, Basel Zbl1186.47001MR2599384
  17. A. Melin, J. Sjöstrand, Determinants of pseudodifferential operators and complex deformations of phase space, Methods Appl. Anal. 9 (2002), 177-237 Zbl1082.35176MR1957486
  18. K. Pravda-Starov, A complete study of the pseudo-spectrum for the rotated harmonic oscillator, J. London Math. Soc., (2) 73 (2006), 745-761 Zbl1106.34060MR2241978
  19. K. Pravda-Starov, Contraction semigroups of elliptic quadratic differential operators, Math. Z. 259 (2008), 363-391 Zbl1139.47033MR2390087
  20. K. Pravda-Starov, Subelliptic estimates for quadratic differential operators, American J. Math. 133 (2011), 39-89 Zbl1257.47058MR2752935
  21. J. Sjöstrand, Parametrices for pseudodifferential operators with multiple characteristics, Ark. för Matematik 12 (1974), 85-130 Zbl0317.35076MR352749
  22. J. Sjöstrand, Singularités analytiques microlocales, 95 (1982), 1-166, Soc. Math. France, Paris Zbl0524.35007MR699623
  23. J. Sjöstrand, Function spaces associated to global I-Lagrangian manifolds, Structure of solutions of differential equations (1996), World Sci. Publ., River Edge, NJ Zbl0889.46027MR1445350
  24. J. Sjöstrand, Resolvent estimates for non-selfadjoint operators via semi-groups, Around the research of Vladimir Maz’ya. III 13 (2010), 359-384, Springer, New York Zbl1198.47068MR2664715
  25. J. Sjöstrand, M. Zworski, Elementary linear algebra for advanced spectral problems, Ann. Inst. Fourier 57 (2007), 2095-2141 Zbl1140.15009MR2394537

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