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For a class of non-selfadjoint $h$--pseudodifferential operators with doublecharacteristics, we give a precise description of the spectrum and establishaccurate semiclassical resolvent estimates in a neighborhood of the origin.Specifically, assuming that the quadratic approximations of the principalsymbol of the operator along the double characteristics enjoy a partialellipticity property along a suitable subspace of the phase space, namely theirsingular space, we give a precise description of the spectrum of the operatorin an ${\cal O}(h)$--neighborhood of the origin. Moreover, when all thesingular spaces are reduced to zero, we establish accurate semiclassicalresolvent estimates of subelliptic type, which depend directly on algebraicproperties of the Hamilton maps associated to the quadratic approximations ofthe principal symbol.

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