A Spectral Transition in the Stability Theory of the 1-D Soler Model

In this talk I will present recent spectral results related to the stability theory of standing waves ψ(t,x) = e^−iωtφ(x) for the one-dimensional Soler model, a nonlinear Dirac equation, with power nonlinearity.

f(s) = s|s|^p−1, p > 0.

The analysis is motivated by the so-called gap property arising in the stability theory of nonlinear Schrödinger equations. More precisely, we study the spectral structure of the Dirac operators obtained by linearization around standing waves, focusing on the existence of eigenvalues inside the spectral gap and resonances at the thresholds of the essential spectrum.

I will describe a sharp spectral transition in the nonlinearity at the critical exponent p = 1: threshold resonances present in the Gross–Neveu case generate eigenvalues inside the spectral gap for 0 < p < 1, whereas no threshold resonances occur for p > 1.