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Nonrelativistic limit of bound-state solutions for nonlinear Dirac equation on noncompact quantum graphs

Guangze Gu, Michael Ruzhansky, Guoyan Wei, Zhipeng Yang

TL;DR

This work analyzes the nonlinear Dirac equation on noncompact quantum graphs, proving the existence of bound-state solutions for frequencies $\omega$ in the relativistic gap and establishing a rigorous nonrelativistic limit as $c \to \infty$ (under small mass) to the nonlinear Schrödinger equation on the same graph. Employing a variational framework for the Dirac operator with Kirchhoff-type vertex conditions, the authors construct critical points of an appropriate action functional, and use concentration-compactness to obtain bound states. In the nonrelativistic regime they show uniform bounds and exponential decay, and prove that the upper spinor component converges to a NLSE bound state while the lower component vanishes at rate $O(1/c_n)$. The results bridge relativistic spinor dynamics on quantum graphs to the well-studied NLSE limit, with precise information on convergence, regularity, and decay that are relevant for applications in nonlinear graph-based quantum systems.

Abstract

In this paper, we investigate the nonrelativistic limit and qualitative properties of bound-state solutions for the nonlinear Dirac equation (NLDE) defined on noncompact quantum graphs: \[ -i c \frac{d}{d x} σ_1 ψ+m c^2 σ_3 ψ-ωψ=g(|ψ|) ψ, \quad \text { in } \mathcal{G} \] where \( g : \mathbb{R}\rightarrow\mathbb{R} \) is a continuous nonlinear function, \( c>0 \) represents the speed of light, \( m>0 \) is the particle's mass, \( ω\in\mathbb{R} \) is related to the frequency, \( σ_1 \) and \( σ_3 \) denote the Pauli matrices, and \(\mathcal{G}\) is a noncompact quantum graph. We establish the existence of bound-state solutions to the NLDE on \(\mathcal{G}\), and prove that these solutions converge toward the corresponding bound-state solutions of a nonlinear Schrödinger equation (NLS) in the nonrelativistic limit (i.e., as the speed of light \( c \to \infty \)) for particles of small mass. Furthermore, we prove uniform boundedness and exponential decay properties of the NLDE solutions, uniformly in \( c \), thereby offering insight into their asymptotic behavior.

Nonrelativistic limit of bound-state solutions for nonlinear Dirac equation on noncompact quantum graphs

TL;DR

This work analyzes the nonlinear Dirac equation on noncompact quantum graphs, proving the existence of bound-state solutions for frequencies in the relativistic gap and establishing a rigorous nonrelativistic limit as (under small mass) to the nonlinear Schrödinger equation on the same graph. Employing a variational framework for the Dirac operator with Kirchhoff-type vertex conditions, the authors construct critical points of an appropriate action functional, and use concentration-compactness to obtain bound states. In the nonrelativistic regime they show uniform bounds and exponential decay, and prove that the upper spinor component converges to a NLSE bound state while the lower component vanishes at rate . The results bridge relativistic spinor dynamics on quantum graphs to the well-studied NLSE limit, with precise information on convergence, regularity, and decay that are relevant for applications in nonlinear graph-based quantum systems.

Abstract

In this paper, we investigate the nonrelativistic limit and qualitative properties of bound-state solutions for the nonlinear Dirac equation (NLDE) defined on noncompact quantum graphs: where is a continuous nonlinear function, represents the speed of light, is the particle's mass, is related to the frequency, and denote the Pauli matrices, and is a noncompact quantum graph. We establish the existence of bound-state solutions to the NLDE on , and prove that these solutions converge toward the corresponding bound-state solutions of a nonlinear Schrödinger equation (NLS) in the nonrelativistic limit (i.e., as the speed of light ) for particles of small mass. Furthermore, we prove uniform boundedness and exponential decay properties of the NLDE solutions, uniformly in , thereby offering insight into their asymptotic behavior.
Paper Structure (11 sections, 20 theorems, 220 equations)

This paper contains 11 sections, 20 theorems, 220 equations.

Table of Contents

  1. Introduction and main results
  2. The functional-analytic setting
  3. Metric graphs and functional setting
  4. The Dirac operator with Kirchhoff-type conditions
  5. The associated quadratic form
  6. The preliminary results
  7. Existence of bound-state solutions of NLDE
  8. Nonrelativistic limit of solutions for NLDE
  9. Uniform boundedness of the bound-state solution for NLDE
  10. Proof of the first part of Theorem \ref{['Thm1.2']}
  11. Proof of the second part of Theorem 1.2

Key Result

Theorem 1.1

Let $\mathcal{G}$ be a noncompact quantum graph and assumptions $(g_1)-(g_5)$ hold with $2<p<\infty$. Then, for every $\omega \in(-mc^2, mc^2 )$, there exists at least one nontrivial bound-state solution $\psi$ of frequency $\omega$ of 1.1.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 13 more