Dirac Operators on Infinite Quantum Graphs

Abstract

We study quantum graphs $\Gamma$ with a finite or countable set $\mathcal{E}$ of edges equipped with the Dirac operators

$$\mathfrak{D}_{\Gamma,Q}=J\frac{d}{dx}+Q,J=\left( \begin{array} [c]{cc} 0 & -1\\ 1 & 0 \end{array} \right) ,$$

where

$$\begin{aligned} \, Q(x) & =\left( \begin{array} [c]{cc} p(x)+r(x) & q(x)\\ q(x) & -p(x)+r(x) \end{array} \right) ,\qquad p,q,r \in L^{\infty}(\Gamma). \end{aligned}$$

We consider the self-adjointness of the unbounded operator $\ \mathcal{D} _{\Gamma,Q,\mathfrak{M}}$ in $L^{2}(\Gamma,\mathbb{C}^{2})$ generated by the Dirac operators $\mathfrak{D}_{\Gamma;Q}$ with domains consisting of spinors

$$u(x)=\left( \begin{array} [c]{c} u^{1}(x)\\ u^{2}(x) \end{array} \right) \in H^{1}(\Gamma,\mathbb{C}^{2}).$$

and with the coupling conditions on the vertices $v\in\mathcal{V}$,

$$\mathfrak{M}u(v)=\mathfrak{a}_{1}(v)u^{1}(v)+\mathfrak{a}_{2}(v)u^{2} (v)=0,v\in\mathcal{V}. \qquad\qquad\qquad\qquad (1)$$

Applying the method of limit operators, we describe the essential spectra of operators $\mathcal{D}_{\Gamma,Q,\mathfrak{M}}$ on the graphs with finite sets of exits to infinity and also on periodic graphs with aperiodic potentials and aperiodic coupling conditions.