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

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.