Spectral analysis of Dirac operators for fermion scattering on topological solitons in the nonlinear $O(3)$ $\sigma$-model

TL;DR

This paper analyzes the spectral properties of Dirac operators in the context of fermion scattering on topological solitons within the nonlinear O(3) sigma-model, focusing on ground states and energy conditions.

Contribution

It provides new criteria for the existence of discrete energy ground states and conditions for non-zero energies of the Dirac operator in this model.

Findings

Existence of discrete positive and negative energy ground states.

A sufficient condition for non-zero energies of the Dirac operator.

Insights into fermion-soliton interactions in the nonlinear sigma-model.

Abstract

We investigate the existence of discrete positive or negative energy ground states of the Dirac operator $H$ which describe the fermion scattering on topological solitons in the nonlinear $O(3)$ $\sigma$-model. Additionally, we provide a sufficient condition to ensure that the positive and negative energies of the Dirac operator $H$ are non-zero.