On spectral stability of the nonlinear Dirac equation

TL;DR

This paper analyzes the spectral stability of solitary waves in the nonlinear Dirac equation across dimensions, showing the absence of embedded eigenvalues beyond thresholds and characterizing eigenvalue behavior near the nonrelativistic limit.

Contribution

It provides new results on the spectral stability by proving the absence of embedded eigenvalues beyond thresholds and describing eigenvalue accumulation in the nonrelativistic limit.

Findings

No embedded eigenvalues beyond thresholds in the essential spectrum.

Eigenvalues with nonzero real part can only emerge from embedded eigenvalues or thresholds.

Eigenvalues accumulate to 0 and ±2mi as ω approaches m.

Abstract

We study the point spectrum of the nonlinear Dirac equation in any spatial dimension, linearized at one of the solitary wave solutions. We prove that, in any dimension, the linearized equation has no embedded eigenvalues in the part of the essential spectrum beyond the embedded thresholds. We then prove that the birth of point eigenvalues with nonzero real part (the ones which lead to linear instability) from the essential spectrum is only possible from the embedded eigenvalues or thresholds, and therefore can not take place beyond the embedded thresholds. We also prove that "in the nonrelativistic limit" $\omega\to m$, the point eigenvalues can only accumulate to $0$ and $\pm 2 m i$.