Browsing by Author "van den Bosch, Hanne"

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    Results on the Spectral Stability of Standing Wave Solutions of the Soler Model in 1-D
    (2023) Aldunate, Danko; Ricaud, Julien; Stockmeyer, Edgardo; van den Bosch, Hanne
    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)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 (phi(0), beta phi(0))C-2 > 0 characterizes groundstates analogously to the Schrodinger case.