Combining geometrical and microlocal tools, this monograph gives detailed proofs of many well/ill-posed results related to the Cauchy problem for diï¬erential operators with non-eï¬ectively hyperbolic double characteristics. Previously scattered over numerous diï¬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 diï¬erential operator P of order m (i. e. one where Pm = dPm = 0) is eï¬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 eï¬ectively hyperbolic, the Cauchy problem for P can be solved for arbitrary lower order terms. If there is a non-eï¬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 insuï¬cient to determine whether the Cauchy problem is well-posed and the behavior of bicharacteristics near the doubly characteristic manifold plays a crucial role. | Cauchy Problem For Differential Operators With Double Characteristics by Tatsuo Nishitani, Paperback | Indigo Chapters