On the spectral stability of the nonlinear Dirac equation with Soler type nonlinearity

Abstract

We study the spectral stability of the nonlinear Dirac operator in di-mension 1 + 1, restricting our attention to nonlinearities of the formf(⟨ψ,βψ⟩C2 )β. We obtain bounds on eigenvalues for the linearized operatoraround standing wave solutions of the form e−iωtϕ0. 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 axesof 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 ⟨ϕ0,βϕ0⟩C2 >0 characterizes ground states analogously to the Schrödinger case.