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The stationary Maxwell-Dirac equations. (English) Zbl 1053.81022

The Maxwell-Dirac equations are the first quantized equations for electronic matter. They consist of a Dirac equation with a Maxwell field term sourced by the Dirac current. This leads to a system of coupled nonlinear partial differential equations, which exhibit phenomena not apparent from perturbation theory or the traditional linear theory. Such a system is stationary if there is a gauge with a time independent spinor \(\phi\), \(\psi = (\exp -iEt) \phi\). It is called isolated if the dependent variables obey some regularity and weak decay conditions. More precisely this means that the component functions belong to some weighted Sobolev space. For stationary and isolated systems the author shows:
{} Such a system has no embedded eigenvalues \(E\) in the essential spectrum, i.e. \(-m\leq E \leq m\).
{} If \(| E| <m\) the corresponding Dirac field decays exponentially.
{} If \(| E| =m\) the system is asymptotically static, i.e. the spatial components of the current decay asymptotically. If the total charge is nonzero the system decays exponentially.
The main methods are weighted Sobolev space methods and regularity results for elliptic partial differential equations.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81V10 Electromagnetic interaction; quantum electrodynamics
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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