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On the microlocal structure of pseudodifferential operators. (English. Russian original) Zbl 0609.35088
Math. USSR, Sb. 56, 515-527 (1987); translation from Mat. Sb., Nov. Ser. 128(170), No. 4, 516-529 (1985).
The authors consider the classical pseudodifferential operators (p.d.o.) on the smooth manifold M. The operators \(\hat H_ 1=H_ 1(x,-i\partial /\partial x)\) and \(\hat H_ 2=H_ 2(x,-i\partial/\partial x)\) are microlocal equivalent at the point \((x_ 0,p_ 0)\in T^*_ 0(M)\) if there exist an elliptic Fourier integral operator \({\hat\Phi}\), associated with some canonical transformation, and an elliptic p.d.o. \(\hat Q\) such that the full symbols of operators \(\hat H_ 2\) and \({\hat\Phi}^{-1}\hat Q\hat H_ 1{\hat \Phi}\) are equal.
Under some assumption the problem of microlocal classification is reduced to the description of orbits of some Lie groups of contact transformations on the space of \((k+1)\)-degree polynomial hamiltonians. The normal forms of hamiltonians are described. Then the equivalents of the (main) symbols are extended to the equivalents of p.d.o.
Reviewer: N.L.Vasilevski
MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
58J40 Pseudodifferential and Fourier integral operators on manifolds
47Gxx Integral, integro-differential, and pseudodifferential operators
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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