Plane wave stability analysis of Hartree and quantum dissipative systems

TL;DR

This paper analyzes the stability of plane wave solutions in quantum systems interacting with complex environments, highlighting the challenges posed by nonlocal potentials and the limitations of asymptotic approximations.

Contribution

It introduces a stability analysis framework for PDE models with nonlocal potentials in quantum dissipative systems, emphasizing Hamiltonian structures and Lyapunov functionals.

Findings

Identification of Hamiltonian structures and Lyapunov functionals for stability analysis

Differences between original models and asymptotic regimes clarified

Perturbative analysis from asymptotic models is insufficient for stability conclusions

Abstract

We investigate the stability of plane wave solutions of equations describing quantum particles interacting with a complex environment. The models take the form of PDE systems with a non local (in space or in space and time) self-consistent potential; such a coupling lead to challenging issues compared to the usual non linear Schr{\"o}dinger equations. The analysis relies on the identification of suitable Hamiltonian structures and Lyapounov functionals. We point out analogies and differences between the original model, involving a coupling with a wave equation, and its asymptotic counterpart obtained in the large wave speed regime. In particular, while the analogies provide interesting intuitions, our analysis shows that it is illusory to obtain results on the former based on a perturbative analysis from the latter.