On gap properties for the linearized 1D Dirac--Soler model

TL;DR

This paper investigates the spectral properties of a linearized Dirac operator in the 1D Soler model, revealing how eigenvalues depend on the nonlinearity power and identifying conditions for the gap property.

Contribution

It establishes the gap property for the linearized Dirac operator in the 1D Soler model for nonlinearities with p ≥ 1 and describes eigenvalue emergence for p < 1.

Findings

Eigenvalues are limited to ground states for p ≥ 1.

Additional eigenvalues appear for p < 1 from the spectrum thresholds.

Thresholds do not admit eigenvalues and have at most one resonance.

Abstract

We study spectral properties of the Dirac operator $L_0$ arising as the upper-right off-diagonal block in the linearization around standing wave solutions of the one-dimensional Soler model with power nonlinearity $f(s)=s|s|^{p-1}$, $p>0$. Our main results concern the so-called gap property: we show that if $p \geq 1$, then the only eigenvalues of $L_0$ are its ground state energies, $-2\omega$ and $0$. In contrast, for $p<1$, additional eigenvalues appear from the thresholds of the essential spectrum. Furthermore, we prove that the thresholds never admit eigenvalues and that they have at most one resonance.