Results on the Spectral Stability of Standing Wave Solutions of the Soler Model in 1-D - Archive ouverte HAL
Article Dans Une Revue Communications in Mathematical Physics Année : 2023

Results on the Spectral Stability of Standing Wave Solutions of the Soler Model in 1-D

Résumé

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 Schrödinger case.

Dates et versions

hal-03388143 , version 1 (20-10-2021)

Identifiants

Citer

Danko Aldunate, Julien Ricaud, Edgardo Stockmeyer, Hanne van den Bosch. Results on the Spectral Stability of Standing Wave Solutions of the Soler Model in 1-D. Communications in Mathematical Physics, 2023, 401 (1), pp.227--273. ⟨10.1007/s00220-023-04646-4⟩. ⟨hal-03388143⟩
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