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Zbl 0657.35102
Bachelot, Alain; Petkov, Vesselin
Existence des opérateurs d'ondes pour les systèmes hyperboliques avec un potentiel périodique en temps. (Existence of the wave operators for the hyperbolic systems with time periodic potential). (French)
[J] Ann. Inst. Henri Poincaré, Phys. Théor. 47, 383-428 (1987). ISSN 0246-0211

We prove the existence of the scattering operator for the wave equation with a potential which is periodic in time and has compact support in space, in dimension greater than or equal to 3, provided the energy is uniformly bounded. The key result is the decay of the local energy. We get strong convergence by using the compactness of the local evolution operator, derived from a microlocal analysis of the propagation of singularities. In the case where the dimension is odd, the decay is exponential for initial data: i) with compact support and ii) included in a subspace of finite codimension. We give some sufficient conditions for the boundedness of the energy by studying the spectrum of the local evolution operator. We extend these results to first order hermitian systems with arbitrary multiplicity and with a periodic potential such as the Dirac system in a periodic electromagnetic field.

MSC 2000:: *35P25 Scattering theory (PDE)
35L10 Second order hyperbolic equations, general
35A27 Sheaf-theoretic methods (PDE)

Keywords: time periodic potential; existence; scattering operator; decay of the local energy; local evolution operator; microlocal analysis; propagation of singularities; spectrum; Dirac system

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