Results on the spectral stability of standing wave solutions of the Soler model in 1-D

Abstract

We study the spectral stability of the nonlinear Dirac operator in dimension $1+1$, restricting our attention to nonlinearities of the form $f(⟨\psi ,\beta \psi {⟩}_{{\mathbb{C}}^2})\beta$. We obtain bounds on eigenvalues for the linearized operator around standing wave solutions of the form $e^{-i\omega t}{\varphi }_0$. For the case of power nonlinearities $f(s)=s|s{|}^{p-1}$, $p>0$, we obtain a range of frequencies $\omega$ 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 $⟨{\varphi }_0,\beta {\varphi }_0{⟩}_{{\mathbb{C}}^2}>0$ characterizes groundstates analogously to the Schrödinger case.

arXiv comment45 pages, 5 figures

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