Affiliations: Université de Cergy Pontoise, 2 av. A. Chauvin, BP 222, 95302 Cergy-Pontoise Cedex, France. E-mails: {Thomas.Duyckaerts,Thierry.Jecko}@u-cergy.fr | Université Paris Est, UFR des Sciences et Technologie, 61, avenue du Général de Gaulle, 94010 Créteil Cedex, France. E-mail: [email protected]
Note: [] Supported by the French ANR grants “OndNonLin” and “ControlFlux”.
Note: [] Supported by the GDR “Maths and Quantum Physics”.
Note: [] Supported by the French ANR grant “NONAa” and by the GDR “Maths and Quantum Physics”.
Abstract: In this article, we analyze the propagation of Wigner measures of a family of solutions to a system of semi-classical pseudodifferential equations presenting eigenvalues crossings on hypersurfaces. We prove the propagation along classical trajectories under a geometric condition which is satisfied, for example, as soon as the Hamiltonian vector fields are transverse or tangent at finite order to the crossing set. We derive resolvent estimates for semi-classical Schrödinger operator with matrix-valued potential under a geometric condition of the same type on the crossing set and we analyze examples of degenerate situations where one can prove transfers between the modes.
Keywords: semi-classical Schrödinger equation, matrix-valued potential, Wigner measure, eigenvalue crossing, normal form
