Degenerated codimension 1 crossings and resolvent estimates.

Croisements dégénérés de codimension 1 et estimations de résolvantes.

Thomas Duyckaerts (1) , Clotilde Fermanian Kammerer (2) , Thierry Jecko (1)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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.

Mots clés: degenerated eigenvalues crossing; Matricial Schrödinger operators; semiclassical resolvent estimates; semiclassical measure; Wigner measure; non-trapping condition; degenerated eigenvalues crossing.

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]

Thomas Duyckaerts, Clotilde Fermanian Kammerer, Thierry Jecko. Degenerated codimension 1 crossings and resolvent estimates.. Asymptotic Analysis, 2009, 65, pp.147-174. ⟨hal-00338331⟩