For a class of non-selfadjoint –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 –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.
Nous décrivons le spectre et établissons des estimations de résolvante semi-classiques dans un voisinage de l’origine pour une classe d’opérateurs -pseudodifférentiels non-autoadjoints à caractéristiques doubles. Plus précisément, sous l’hypothèse que les approximations quadratiques du symbole principal de l’opérateur sont elliptiques sur un sous-espace particulier de l’espace des phases, dénommé espace singulier, nous donnons une description précise du spectre de cet opérateur dans un -voisinage de l’origine. De plus, lorsque tous les espaces singuliers sont nuls, nous établissons des estimations de résolvante semi-classiques de type sous-elliptique qui dépendent directement de propriétés algébriques des applications hamiltoniennes des approximations quadratiques du symbole principal.
Keywords: non-selfadjoint operator, eigenvalue, resolvent estimate, subelliptic estimates, double characteristics, singular space, pseudodifferential calculus, Wick calculus, FBI transform, Grushin problem
Mot clés : Opérateurs non-autoadjoints, Valeurs propres, Estimations de résolvante, Estimations sous-elliptiques, Caractéristiques doubles, Espace singulier, Calcul pseudo-différentiel, Calcul de Wick, Transformation FBI, Problème de Grushin
@article{AIF_2013__63_3_985_0, author = {Hitrik, Michael and Pravda-Starov, Karel}, title = {Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics}, journal = {Annales de l'Institut Fourier}, pages = {985--1032}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {63}, number = {3}, year = {2013}, doi = {10.5802/aif.2782}, zbl = {1292.35185}, mrnumber = {3137478}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2782/} }
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 SP - 985 EP - 1032 VL - 63 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2782/ DO - 10.5802/aif.2782 LA - en ID - AIF_2013__63_3_985_0 ER -
%0 Journal Article %A Hitrik, Michael %A Pravda-Starov, Karel %T Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics %J Annales de l'Institut Fourier %D 2013 %P 985-1032 %V 63 %N 3 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2782/ %R 10.5802/aif.2782 %G en %F AIF_2013__63_3_985_0
Hitrik, Michael; Pravda-Starov, Karel. Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics. Annales de l'Institut Fourier, Volume 63 (2013) no. 3, pp. 985-1032. doi : 10.5802/aif.2782. http://archive.numdam.org/articles/10.5802/aif.2782/
[1] Loi de Weyl presque sûre et résolvante pour des opérateurs différentiels non-autoadjoints, CMLS, École Polytechnique (2008) (Ph.D. thesis)
[2] Pseudo-spectra of semiclassical (pseudo)differential operators, Comm. Pure Appl. Math., Volume 57 (2004), pp. 384-415 | DOI | MR | Zbl
[3] Spectral asymptotics in the semi-classical limit, Cambridge University Press, 1999 | MR | Zbl
[4] Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators, Math. Annalen, Volume 342 (2008), pp. 177-243 | DOI | MR | Zbl
[5] Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten laplacians, SLN, 1862, Springer Verlag, 2005 | MR | Zbl
[6] Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum, Mem. Soc. Math. France (N.S.), Volume 39 (1989), pp. 1-124 | Numdam | MR | Zbl
[7] Spectra and semigroup smoothing for non-elliptic quadratic operators, Math. Ann., Volume 334 (2009), pp. 801-846 | DOI | MR | Zbl
[8] Semiclassical hypoelliptic estimates for non-selfadjoint operators with double characteristics, Comm. P.D.E., Volume 35 (2010), pp. 988-1028 | DOI | MR | Zbl
[9] Tunnel effect for Kramers-Fokker-Planck type operators, Ann. Henri Poincaré, Volume 9 (2008), pp. 209-274 | DOI | MR | Zbl
[10] Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high degree potential, Arch. Ration. Mech. Anal., Volume 171 (2004), pp. 151-218 | DOI | MR | Zbl
[11] Semiclassical analysis for the Kramers-Fokker-Planck equation, Comm. PDE, Volume 30 (2005), pp. 689-760 | DOI | MR | Zbl
[12] The analysis of linear partial differential operators, I–IV, Springer-Verlag, 1985 | Zbl
[13] Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., Volume 219 (1995), pp. 413-449 | DOI | MR | Zbl
[14] The Wick calculus of pseudodifferential operators and some of its applications, Cubo Mat. Educ., Volume 5 (2003), pp. 213-236 | MR
[15] Some Facts About the Wick Calculus, Pseudo-differential operators. Quantization and signals (Lecture Notes in Mathematics), Volume 1949, Springer-Verlag, Berlin, 2008, pp. 135-174 | MR | Zbl
[16] Metrics on the phase space and non-selfadjoint pseudo-differential operators, Pseudo-Differential Operators. Theory and Applications, 3, Birkhäuser Verlag, Basel, 2010 | MR | Zbl
[17] Determinants of pseudodifferential operators and complex deformations of phase space, Methods Appl. Anal., Volume 9 (2002), pp. 177-237 | MR | Zbl
[18] A complete study of the pseudo-spectrum for the rotated harmonic oscillator, J. London Math. Soc., (2), Volume 73 (2006) no. 3, pp. 745-761 | DOI | MR | Zbl
[19] Contraction semigroups of elliptic quadratic differential operators, Math. Z., Volume 259 (2008), pp. 363-391 | DOI | MR | Zbl
[20] Subelliptic estimates for quadratic differential operators, American J. Math., Volume 133 (2011), pp. 39-89 | DOI | MR | Zbl
[21] Parametrices for pseudodifferential operators with multiple characteristics, Ark. för Matematik, Volume 12 (1974), pp. 85-130 | DOI | MR | Zbl
[22] Singularités analytiques microlocales (Astérisque), Volume 95, Soc. Math. France, Paris, 1982, pp. 1-166 | Numdam | MR | Zbl
[23] Function spaces associated to global I-Lagrangian manifolds, Structure of solutions of differential equations (1996) | MR | Zbl
[24] Resolvent estimates for non-selfadjoint operators via semi-groups, Around the research of Vladimir Maz’ya. III (Int. Math. Ser.), Volume 13, Springer, New York, 2010, pp. 359-384 | MR | Zbl
[25] Elementary linear algebra for advanced spectral problems, Ann. Inst. Fourier, Volume 57 (2007), pp. 2095-2141 | DOI | Numdam | MR | Zbl
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