arXiv subjects

Lysianne Hari

Publications and source records attributed to Lysianne Hari.

5 recordsLinked to original sources

arXiv2021

Propagation of Coherent States through Conical Intersections

In this paper, we analyze the propagation of a wave packet through a conical intersection. This question has been addressed for Gaussian wave packets in the 90s by George Hagedorn and we consider here a more general setting. We focus on the case of Schr{ö}dinger equation but our methods are general enough to be adapted to systems presenting codimension 2 crossings and to codimension 3 ones with specific geometric conditions. Our main Theorem gives explicit transition formulas for the profiles when passing through a conical crossing point, including precise computation of the transformation of the phase. Its proof is based on a normal form approach combined with the use of superadiabatic projectors and the analysis of their degeneracy close to the crossing.

math.AP↗

arXiv2017

Large data scattering for the defocusing NLKG on waveguide $\mathbb R^d\times\mathbb T$

We consider the pure-power defocusing nonlinear Klein-Gordon equation, in the energy subcritical case, posed on the product space $\mathbb R^d\times \mathbb T$, where $\mathbb T$ is the one-dimensional flat torus. In this framework, we prove that scattering holds for any initial data belonging to the energy space $H^1 \times L^2$ for $1\leq d\leq 4$. The strategy consists in proving a suitable profile decomposition theorem in $\mathbb R^d\times \mathbb T$ to pursue a concentration-compactness \& rigidity method.

math.AP↗

arXiv2016

Small data scattering for energy-subcritical and critical Nonlinear Klein Gordon equations on product spaces

We study small data scattering of solutions to Nonlinear Klein-Gordon equations with suitable pure power nonlinearities, posed on $\mathbb{R}^d\times \mathcal{M}^k$ with $k\leq2$ and $d\geq1$ and $\mathcal{M}^k$ a compact Riemannian manifold. As a special case we cover the $H^1-$critical NLKG on $\mathbb{R} \times M^2$.

math.AP↗

arXiv2014

Nonlinear Propagation of Coherent States through Avoided Energy Level Crossing

We study the propagation of wave packets for a one-dimensional system of two coupled Schrödinger equations with a cubic nonlinearity, in the semi-classical limit. Couplings are induced by the nonlinearity and by the potential, whose eigenvalues present an "avoided crossing": at one given point, the gap between them reduces as the semi-classical parameter becomes smaller. For data which are coherent states polarized along an eigenvector of the potential, we prove that when the wave function propagates through the avoided crossing point, there are transitions between the eigenspaces at leading order. We analyze the nonlinear effects, which are noticeable away from the crossing point, but see that in a small time interval around this point, the nonlinearity's role is negligible at leading order, and the transition probabilities can be computed with the linear Landau-Zener formula.

math-ph↗

arXiv2012

Coherent states for systems of $L^2-$supercritical nonlinear Schrödinger equations

We consider the propagation of wave packets for a nonlinear Schrödinger equation, with a matrix-valued potential, in the semi-classical limit. For a matrix-valued potential, Strichartz estimates are available under long range assumptions. Under these assumptions, for an initial coherent state polarized along an eigenvector, we prove that the wave function remains in the same eigenspace, in a scaling such that nonlinear effects cannot be neglected. We also prove a nonlinear superposition principle for these nonlinear wave packets.

math.AP↗