Summary
We consider evolution equations ∂u/∂t = ia w(x, D)u where a is the (real valued) Weyl symbol of the operator A = a w. For instance, Schrödinger-like equations. After recalling what are generalized Fourier integral operators in the framework of the Weyl-Hörmander calculus, we give conditions on a and on the dynamics of its hamiltonian flow which imply: 1. The operator a w is essentially self-adjoint and the propagators e itA are bounded between (conveniently related) generalized Sobolev spaces. 2. The propagators e itA are generalized Fourier integral operators.
2000 AMS Subject Classification: 35G10, 35S05, 35S30
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References
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Dedicato a Ferruccio Colombini in occasione del suo sessantesimo compleanno
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Bony, JM. (2009). Evolution Equations and Generalized Fourier Integral Operators. In: Bove, A., Del Santo, D., Murthy, M. (eds) Advances in Phase Space Analysis of Partial Differential Equations. Progress in Nonlinear Differential Equations and Their Applications, vol 78. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4861-9_4
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DOI: https://doi.org/10.1007/978-0-8176-4861-9_4
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