Results on the Spectral Stability of Standing Wave Solutions of the Soler Model in 1-D.

We study the spectral stability of the nonlinear Dirac operator in dimension 1 + 1 , restricting our attention to nonlinearities of the form f (ψ , β ψ C 2) β . We obtain bounds on eigenvalues for the linearized operator around standing wave solutions of the form e - i ω t ϕ 0 . For the case of power nonlinearities f (s) = s | s | p - 1 , p > 0 , we obtain a range of frequencies ω such that the linearized operator has no unstable eigenvalues on the axes of the complex plane. As a crucial part of the proofs, we obtain a detailed description of the spectra of the self-adjoint blocks in the linearized operator. In particular, we show that the condition ϕ 0 , β ϕ 0 C 2 > 0 characterizes groundstates analogously to the Schrödinger case. [ABSTRACT FROM AUTHOR]