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

Hitrik, Michael; Pravda-Starov, Karel

Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 985-1032.

Nous décrivons le spectre et établissons des estimations de résolvante semi-classiques dans un voisinage de l’origine pour une classe d’opérateurs $h$-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 $𝒪\left(h\right)$-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.

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 $𝒪\left(h\right)$–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.

Classification : 35H20, 35P20, 35S05, 47A10, 47B44
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

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