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We consider the nonlinear Dirac equation with Soler-type nonlinearityconcentrated at one point and present a detailed study of the spectrum oflinearization at solitary waves. We then consider two different perturbationsof the nonlinearity which break the $\mathbf{SU}(1,1)$-symmetry: the firstpreserving and the second breaking the parity symmetry. We show that aperturbation which breaks the $\mathbf{SU}(1,1)$-symmetry but not the paritysymmetry also preserves the spectral stability of solitary waves. Then weconsider a perturbation which breaks both the $\mathbf{SU}(1,1)$-symmetry andthe parity symmetry and show that this perturbation destroys the stability ofweakly relativistic solitary waves. The developing instability is due to thebifurcations of positive-real-part eigenvalues from the embedded eigenvalues$\pm 2\omega\mathrm{i}$.

Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity