Vincent Louatron | Searcharxiv

arXiv2025

A microlocal Cauchy problem through a crossing point of Hamiltonian flows

In this paper, we consider $2\times 2$ matrix-valued pseudodifferential equations in which the two characteristic sets intersect with finite contact order. We show that the asymptotic behavior of its solution changes dramatically before and after the crossing point, and provide a precise asymptotic formula. This is a generalization of the previous results for matrix-valued Schr\"odinger operators and Landau-Zener models. The proof relies on a normal form reduction and a detailed analysis of a simple first-order system.

math.AP↗

arXiv2024

Semiclassical resonances for matrix Schrödinger operators with vanishing interactions at crossings of classical trajectories

We study the semiclassical distribution of resonances of a $2 \times 2$ matrix Schrödinger operator, obtained as a reduction of an Hamiltonian when studying molecular predissociation models under the Born-Oppenheimer approximation. The energy considered is above the energy-level crossing of the two associated classical trajectories, and is respectively trapping and non-trapping for those trajectories. Under a condition between the contact order $m$ of the crossings and the vanishing order $k$ of the interaction term at the crossing points, we show that, asymptotically in the semiclassical limit $h \to 0^+$, the imaginary part of the resonances is of size $h^{1+2(k+1)/(m+1)}$ in the general case and shrinks to $h^{1+2(k+2)/(m+1)}$ when both $k$ and $m$ are odd. We also compute the first term of the associated asymptotic expansions.

math-ph↗