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

Abstract

By using the Wigner transform, it is shown that the nonlinear Schr$\ddot{\text{o}}$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$\ddot{\text{o}}$dinger equation under the quantum stability assumption.

21pages. arXiv admin note: text overlap with arXiv:1810.10955, arXiv:1807.05254; text overlap with arXiv:quant-ph/0505004 by other authors