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

Autor: Aldunate, Danko, Ricaud, Julien, Stockmeyer, Edgardo, Bosch, Hanne Van Den
Rok vydání: 2021
Předmět:
Zdroj: Communications in Mathematical Physics 401 (2023) 227--273
Druh dokumentu: Working Paper
DOI: 10.1007/s00220-023-04646-4
Popis: We study the spectral stability of the nonlinear Dirac operator in dimension $1+1$, restricting our attention to nonlinearities of the form $f(\langle\psi,\beta \psi\rangle_{\mathbb{C}^2}) \beta$. We obtain bounds on eigenvalues for the linearized operator around standing wave solutions of the form $e^{-i\omega t} \phi_0$. 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 $\langle\phi_0,\beta \phi_0\rangle_{\mathbb{C}^2} > 0$ characterizes groundstates analogously to the Schr\"odinger case.
Comment: 45 pages, 5 figures
Databáze: arXiv
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