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

Daiju Funakawa, Satoshi Okumura, Yuki Ueda

TL;DR

This work analyzes the spectral properties of Dirac operators describing fermion scattering on topological solitons in the nonlinear $O(3)$ sigma-model. By reformulating the problem in polar coordinates, imposing a hedgehog ansatz, and decomposing the Hilbert space into grand-spin sectors, the authors derive radial operators whose spectra control the existence of discrete ground states. They establish a concrete variational condition that yields discrete ground states and provide a mass-parameter regime ensuring nonzero positive and negative ground energies, including an explicit hedgehog-based example. The results illuminate bound-state formation in fermion-soliton systems and identify parameter regimes with robust spectral gaps, contributing to rigorous understanding of fermion-soliton interactions in field theories.

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 $O(3)$ $σ$-model. Additionally, we provide a sufficient condition to ensure that the positive and negative energies of the Dirac operator $H$ are non-zero.

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

TL;DR

This work analyzes the spectral properties of Dirac operators describing fermion scattering on topological solitons in the nonlinear sigma-model. By reformulating the problem in polar coordinates, imposing a hedgehog ansatz, and decomposing the Hilbert space into grand-spin sectors, the authors derive radial operators whose spectra control the existence of discrete ground states. They establish a concrete variational condition that yields discrete ground states and provide a mass-parameter regime ensuring nonzero positive and negative ground energies, including an explicit hedgehog-based example. The results illuminate bound-state formation in fermion-soliton systems and identify parameter regimes with robust spectral gaps, contributing to rigorous understanding of fermion-soliton interactions in field theories.

Abstract

We investigate the existence of discrete positive or negative energy ground states of the Dirac operator which describe the fermion scattering on topological solitons in the nonlinear -model. Additionally, we provide a sufficient condition to ensure that the positive and negative energies of the Dirac operator are non-zero.
Paper Structure (8 sections, 13 theorems, 73 equations)

This paper contains 8 sections, 13 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Discrete and essential spectrum
  3. Dirac operators with hedgehog ansatz
  4. Differential operators under the change of coordinates
  5. Hedgehog ansatz
  6. Reduced part with respect to the grand spin
  7. Discrete ground states
  8. Non-zero positive and negative energies

Key Result

Theorem 1.4

Let us consider a profile function $F$. If there exist $\ell \in \mathbb{Z}$ and $t\in \{1,-1\}$ such that then $H({\bm{n}}(F))$ has a discrete positive or negative energy ground state, where $C_0^\infty((0,\infty))$ is the set of all compactly supported $C^\infty$-functions on $(0,\infty)$ and for all $u,v\in L^2((0,\infty), r{\rm d} r)$.

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • ...and 17 more