The Cauchy problem for hyperbolic operators

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The goal of this book is a construction of the fundamental solution to the Cauchy problem for hyperbolic operators with multiple characteristics.Well-posedness of the problem in various functional spaces as well as a propagation of singularities of the solutions are investigated, too.For operators with multiple characteristics so called Levi conditions play a crucial rule. Levi conditions described in the book allow to construct fundamental solutions.Starting point of the treatment is a turning point theory for ordinary differential equations. An approach is given which is available to the turning points of infinite and higher order equations, too. Applications to the problem for partial differential equations (Cauchy problem, local solvability and hypoellipticity) with multiple characteristics and to some problems of quantum mechanics are given.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 coefficients. This approach needs Fourier integral operators with inhomogeneous phase functions. Necessary knowledge on these ones is given, too.