Solvability and complex limit bicharacteristics

We shall study the solvability of pseudodifferential operators which are not of principal type. The operator will have complex principal symbol satisfying condition (Ψ) and we shall consider the limits of semibicharacteristics at the set where the principal symbol vanishes of at least second order. The convergence shall be as smooth curves, and we shall assume that the normalized complex Hamilton vector field of the principal symbol over the semicharacteristics converges to a real vector field. Also, we shall assume that the linearization of the real part of the normalized Hamilton vector field at the semibicharacteristic is tangent to and bounded on the tangent space of a Lagrangean submanifold at the semibicharacteristics, which we call a grazing Lagrangean space. Under these conditions one can invariantly define the imaginary part of the subprincipal symbol. If the quotient of the imaginary part of the subprincipal symbol with the norm of the Hamilton vector field switches sign from − to + on the bicharacteristics and becomes unbounded as they converge to the limit, then the operator is not solvable at the limit bicharacteristic.