We consider the nonlinear Dirac equation with Soler-type nonlinearity concentrated at one point and present a detailed study of the spectrum of linearization at solitary waves. We then consider two different perturbations of the nonlinearity which break the $ \mathbf{SU}(1,1) $ symmetry: the first preserving and the second breaking the parity symmetry. We show that a particular perturbation which breaks the $ \mathbf{SU}(1,1) $ symmetry but not the parity symmetry also preserves the spectral stability of solitary waves. Then we consider a particular perturbation which breaks both the $ \mathbf{SU}(1,1) $ symmetry and the parity symmetry and show that this perturbation destroys the stability of weakly relativistic solitary waves. This instability is due to the bifurcations of positive-real-part eigenvalues from the embedded eigenvalues $ \pm 2\omega \mathrm{i} $.
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Figure 1. Location of eigenvalues of the linearized operator $ \mathbf{A}(\omega,\kappa) $ for different values of parameters $ \omega\in(-m,m) $ and $ \kappa\in \mathbb{R} $. In the while area, there are no eigenvalues besides $ \lambda = 0 $ and $ \lambda = \pm 2\omega \mathrm{i} $. The virtual level curves $ \omega = \mathcal{T}_\kappa $ (dotted curves for $ \kappa<2^{-1/2}-1 $ and $ \kappa>2^{-1/2} $) correspond to virtual levels at thresholds $ \pm \mathrm{i}(m-\vert {\omega} \vert) $. The lightly shaded regions between the virtual level curves and the Kolokolov curves $ \omega = \Omega_\kappa $ (solid curves for $ \kappa<-1/3 $ and $ \kappa>1 $) correspond to two purely imaginary eigenvalues in the spectral gap. The Kolokolov curves correspond to the collision of two eigenvalues at $ \lambda = 0 $ (as indicated by the Kolokolov condition). On the other side of the Kolokolov curves, in the crosshatched regions on the plot, there is a pair of real eigenvalues $ \pm\lambda\in \mathbb{R} $ (linear instability). In these regions, one has $ \pm\lambda\to\pm\infty $ as $ \omega $ and $ \kappa $ approach the curve $ \omega = 2\Omega_\kappa $ (dash-dot curve for $ \kappa<-1/2 $). At $ \kappa = -1 $, $ \omega = 0 $ (point of intersection of the three curves), the spectrum of the linearized operator $ \mathbf{A} $ consists of the whole complex plane
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Location of eigenvalues of the linearized operator