Nonlinear Dirac equations on noncompact quantum graphs with potentials: Multiplicity and Concentration

In this paper, we study the existence and multiplicity of solutions to the following class of nonlinear Dirac equations (NLDE) on noncompact quantum graphs: $$-i\,\varepsilon c\,\sigma_1\,\partial_x u + m c^2 \sigma_3 u + V(x)\,u = f(|u|)\,u, \quad x\in \mathcal{G}, \tag{P}$$ where $V:\mathcal{G}\to\mathbb{R}$ and $f:\mathbb{R}\to\mathbb{R}$ are continuous, $\varepsilon>0$ is a semiclassical parameter, $m>0$ denotes the mass, and $c>0$ the speed of light. Here $\sigma_1,\sigma_3$ are Pauli matrices, and $\mathcal{G}$ is a noncompact quantum graph. We prove that when $\varepsilon$ is sufficiently small, the number of solutions to $(P)$ is at least the number of global minima of $V$. Moreover, these solutions exhibit semiclassical concentration: as $\varepsilon\to0$, their concentration points approach the set of global minima of $V$.