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Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics. (English. French summary) Zbl 1292.35185
Summary: 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.

##### MSC:
 35P20 Asymptotic distributions of eigenvalues in context of PDEs 35S05 Pseudodifferential operators as generalizations of partial differential operators 35H20 Subelliptic equations 47A10 Spectrum, resolvent 47G30 Pseudodifferential operators 47B44 Linear accretive operators, dissipative operators, etc.
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