Degenerated codimension 1 crossings and resolvent estimates

TL;DR

This paper studies the behavior of solutions to semi-classical pseudodifferential equations with eigenvalue crossings, establishing propagation of Wigner measures and resolvent estimates under geometric conditions, and exploring mode transfer in degenerate cases.

Contribution

It introduces new propagation results for Wigner measures and resolvent estimates in systems with eigenvalue crossings, including degenerate situations with mode transfer.

Findings

Propagation of Wigner measures along classical trajectories under geometric conditions

Derivation of resolvent estimates for semi-classical Schrödinger operators with matrix potentials

Analysis of degenerate crossing scenarios with mode transfer mechanisms

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 Schr\"odinger 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.