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Evolution equations and microlocal analysis. (English) Zbl 1055.35010

Colombini, Ferruccio (ed.) et al., Hyperbolic problems and related topics. Proceedings of the conference, Cortona, Italy, September 10–14, 2002. Somerville, MA: International Press (ISBN 1-57146-150-7/pbk). Grad. Ser. Anal., 17-40 (2003).
The author considers such a basic problem: given a self-adjoint unbounded operator \(A\) on \(L^2(\mathbb{R}^n)\), and \(A\) is a differential or pseudodifferential operator with symbol \(a(x,\xi)\), what are the properties of the operator \(P_t= \exp\{itA\}\) others than to be a one parameter group of unitary operators? By using Weyl-Hörmander calculus he mainly obtains the result in this paper as follows:
Given a metric \(g_0\) and a canonical transformation \(F\), calling \(g_1\) the direct image of \(g_0\) by \(F\), one can define a class of operators denoted by \(\text{FIO}(F,g_0,g_1)\) such that the two properties are satisfied: 1. There is a good symbolic calculus for the composition of these operators. 2. The conjugate of a \(g_0\)-pseudo-differential operator by an invertible element of \(\text{FIO}(F,g_0,g_1)\) is a \(g_1\)-pseudo-differential operator.
Besides, the author partially answered the question: Given an operator \(A\) with symbol \(a\) and a metric \(g_0\), under what conditions one can find canonical transformations \(F_t\) and metrics \(g_t\) such that \(P_t\) belongs to \(\text{FIO}(F_t,g_0,g_t)\)? The necessary condition on the symbol \(a\) is obtained in this paper, but the sufficient condition will be given in the author’s subsequent papers.
For the entire collection see [Zbl 1035.35001].

MSC:

35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35S05 Pseudodifferential operators as generalizations of partial differential operators
35S30 Fourier integral operators applied to PDEs
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