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

TL;DR

This paper studies the behavior of bound-state solutions for the nonlinear Dirac equation on noncompact quantum graphs, showing their convergence to nonlinear Schrödinger solutions as the speed of light increases, with uniform bounds and decay properties.

Contribution

It establishes the existence of bound states for NLDE on quantum graphs and proves their convergence to NLS solutions in the nonrelativistic limit, including uniform boundedness and decay results.

Findings

Bound-state solutions exist for NLDE on noncompact quantum graphs.

Solutions converge to NLS bound states as c → ∞.

Solutions are uniformly bounded and decay exponentially, uniformly in c.

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: \[ -i c \frac{d}{d x} \sigma_1 \psi+m c^2 \sigma_3 \psi-\omega \psi=g(|\psi|) \psi, \quad \text { in } \mathcal{G} \] 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, \( \omega\in\mathbb{R} \) is related to the frequency, \( \sigma_1 \) and \( \sigma_3 \) 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 Schr\"odinger 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.