Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity

We study the point spectrum of the linearization at a solitary wave solution phi(omega)(x)e(-i omega t) t to the nonlinear Dirac equation in R-n, for all n >= 1, with the nonlinear term given by f (psi* beta psi)beta psi (known as the Soler model). We focus on the spectral stability, that is, the absence of eigenvalues with positive real part, in the non-relativistic limit omega -> m - 0, in the case when f is an element of C-1 (R \textbackslash {0}), f (tau) = vertical bar tau vertical bar(k) + O (vertical bar tau vertical bar(K)) for tau -> 0, with 0 < k < K. For n >= 1, we prove the spectral stability of small amplitude solitary waves (omega less than or similar to m) for the charge-subcritical cases K less than or similar to /n (in particular, 1 < k <= 2 when n = 1) and for the ``charge-critical case'' k = 2/n (with K > 4/n). An important part of the stability analysis is the proof of the absence of bifurcation of nonzero-real-part eigenvalues from the embedded threshold points at +/- 2mi. Our approach is based on constructing a new family of exact bi-frequency solitary wave solutions in the Soler model and on the analysis of the behavior of ``nonlinear eigenvalues'' (characteristic roots of holomorphic operator-valued functions). (C) 2019 Elsevier Inc. All rights reserved.