On global attraction to quantum stationary states. Dirac equation with mean field interaction

TL;DR

This paper proves that solutions to a nonlinear Dirac equation with mean field interaction in three dimensions tend to nonlinear eigenstates over time, demonstrating a form of global attraction driven by energy transfer and radiation.

Contribution

It establishes the long-time convergence of solutions to nonlinear eigenstates in a U(1)-invariant Dirac model, linking quantum stationary states to nonlinear eigenfunctions.

Findings

Solutions converge to nonlinear eigenfunctions as time approaches infinity.

Energy transfer from lower harmonics to the continuous spectrum causes dispersive radiation.

Global attraction occurs under generic assumptions.

Abstract

We consider a U(1)-invariant nonlinear Dirac equation in dimension $n=3$, interacting with itself via the mean field mechanism. We analyze the long-time asymptotics of solutions and prove that, under certain generic assumptions, each finite charge solution converges as $t\to\pm\infty$ to the two-dimensional set of all "nonlinear eigenfunctions" of the form $\phi(x)e\sp{-i\omega t}$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. The research is inspired by Bohr's postulate on quantum transitions and Schr\"odinger's identification of the quantum stationary states to the nonlinear eigenfunctions of the coupled U(1)-invariant Maxwell-Schr\"odinger and Maxwell-Dirac equations.