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Existence of excited states for a nonlinear Dirac field. (English) Zbl 0696.35158
Summary: We prove the existence of infinitely many stationary states for the following nonlinear Dirac equation $i\gamma^{\mu}\partial_{\mu}\psi -m\psi +({\bar \psi}\psi)\psi =0.$ Seeking for eigenfunctions splitted in spherical coordinates leads us to analyze a nonautonomous dynamical system in $${\mathbb{R}}^ 2$$. The number of eigenfunctions is given by the number of intersections of the stable manifold of the origin with the curve of admissible datum. This proves the existence of infinitely many stationary states, ordered by the number of nodes of each component.

##### MSC:
 35Q99 Partial differential equations of mathematical physics and other areas of application 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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##### References:
 [1] Cazenave, T., Vazquez, L.: Existence of localized solutions for a classical nonlinear Dirac field. Commun. Math. Phys.105, 35-47 (1986) · Zbl 0596.35117 [2] Jones, C., K?pper, T.: On the infinitely many solutions of a semilinear elliptic equation. SIAM J. Math. Anal.17, 803-835 (1986) · Zbl 0606.35032 [3] Merle, F.: Existence of stationary states for nonlinear Dirac equations. J. Differ. Eq. (to appear) · Zbl 0696.35154 [4] Soler, M.: Classical, stable nonlinear spinor field with positive rest energy. Phys. Rev. D1, 2766-2769 (1970) [5] Vazquez, L.: Localized solutions of a nonlinear spinor field. J. Phys. A10, 1361-1368 (1977) [6] Vazquez, L.: Personal communication [7] Wakano, M.: Intensely localized solutions of the classical Dirac-Maxwell field equations. Prog. Theor. Phys. (Kyoto)35, 1117-1141 (1966) [8] Hubbart, J.H., West, B.H.: MacMath. Ithaca, London: Cornell University 1985
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