The perturbational stability of the Schr$\ddot{o}$dinger equation

TL;DR

This paper demonstrates quantum Landau damping for the Wigner-Poisson system and proves the existence and stability of solutions to the nonlinear Schrödinger equation using phase space analysis.

Contribution

It introduces a phase space approach to quantum nonlinear Schrödinger equations and establishes quantum Landau damping and stability results.

Findings

Quantum Landau damping exists for the Wigner-Poisson system.

Existence and stability of the nonlinear Schrödinger equation are proven.

Phase space description aligns quantum and classical plasma theories.

Abstract

By using the Wigner transform, it is shown that the nonlinear Schr$\ddot{\textmd{o}}$dinger equation can be described, in phase space, by a kinetic theory similar to the Vlasov equation which is used for describing a classical collisionless plasma. In this paper we mainly show Landau damping in the quantum sense, namely,quantum Landau damping exists for the Wigner-Poisson system. At the same time, we also prove the existence and the stability of the nonlinear Schr$\ddot{\textmd{o}}$dinger equation under the quantum stability assumption.