Combining geometrical and microlocal tools this monograph gives detailed proofs of many well
ill-posed results related to the Cauchy problem for di erential operators with non-e ectively
hyperbolic double characteristics. Previously scattered over numerous di erent publications
the results are presented from the viewpoint that the Hamilton map and the geometry of
bicharacteristics completely characterizes the well ill-posedness of the Cauchy problem. A
doubly characteristic point of a di erential operator P of order m (i.e. one where Pm = dPm =
0) is e ectively hyperbolic if the Hamilton map FPm has real non-zero eigen values. When the
characteristics are at most double and every double characteristic is e ectively hyperbolic
the Cauchy problem for P can be solved for arbitrary lower order terms. If there is a non-e
ectively hyperbolic characteristic solvability requires the subprincipal symbol of P to lie
between -Pµj and Pµj where iµj are the positive imaginary eigenvalues of FPm . Moreover if 0
is an eigenvalue of FPm with corresponding 4 × 4 Jordan block the spectral structure of FPm is
insu cient to determine whether the Cauchy problem is well-posed and the behavior of
bicharacteristics near the doubly characteristic manifold plays a crucial role.
