The Cauchy Problem for Hyperbolic Operators. Multiple Characteristics.Micro-Local Approach

A construction of the fundamental solution to the Cauchy problem for hyperbolic operators with multiple characteristics is the target of this book. Investigations of the problem in various functional spaces and a propagation of singularities of the solutions are also presented. For operators with multiple characteristics, so-called Levy conditions play a crucial rule. Levy conditions described in the book allow the construction of fundamental solutions. A turning point theory for ordinary differential equations is the starting area of the treatment. An approach is also given which is available to the turning points of infinite and higher order equations. Applications to the problem are given for partial differential equations (Cauchy problem, local solvability and hypoellipticity) with multiple characteristics and to some problems of quantum mechanics. The approach represented in the book is essentially based on the zeros of the complete symbol of the operator. For operators with variable coefficients, hyperbolicity conditions are formulated by means of these zeros similarly to Hadamard's conditions for operators with constant coefficient.
This approach needs Fourier integral operators with inhomogeneous phase functions. The required knowledge on these is given.