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On global attraction to quantum stationary states. Dirac equation with mean field interaction. (English) Zbl 1211.35053

Summary: We consider a \(\mathbb{U}(1)\)-invariant nonlinear Dirac equation, 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 \(\phi(x)e^{-i\omega t}\). 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 Schrödinger’s identification of the quantum stationary states to the nonlinear eigenfunctions of the coupled \(\mathbb{U}(1)\)-invariant Maxwell-Schrödinger and Maxwell-Dirac equations.

MSC:

35B41 Attractors
35Q41 Time-dependent Schrödinger equations and Dirac equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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