Two-Dimensional Dirac Operators with Interactions on Unbounded Smooth Curves

Abstract

We consider the 2D Dirac operator with singular potentials

$$\mathfrak{D}_{\boldsymbol{A},\Phi,Q_{\sin}}\boldsymbol{u}(x)=\left( \mathfrak{D}_{\boldsymbol{A},\Phi}+Q_{\sin}\right) \boldsymbol{u} (x),\quad x\in\mathbb{R}^{2}, $$

(1)

where

$$\mathfrak{D}_{\boldsymbol{a},\Phi}= {\displaystyle\sum\limits_{j=1}^{2}} \sigma_{j}\left( i\partial_{x_{j}}+a_{j}\right) +\sigma_{3}m+\Phi I_{2}; $$

(2)

here $\sigma_{j},j=1,2,3,$ are Pauli matrices, $\boldsymbol{a=}(a_{1},a_{2})$ is the magnetic potential with $a_{j}\in L^{\infty}(\mathbb{R}^{2}),\Phi\in L^{\infty}(\mathbb{R)}$ is the electrostatic potential, $Q_{\sin} =Q\delta_{\Gamma}$ is the singular potential with the strength matrix $Q=\left( Q_{ij}\right)_{i,j=1}^{2}$, and $\delta_{\Gamma}$ is the delta-function with support on a $C^{2}-$curve $\Gamma$, which is the common boundary of the domains $\Omega_{\pm}\subset\mathbb{R}^{2}.$ We associate with the formal Dirac operator $\mathfrak{D}_{\boldsymbol{a},\Phi,Q_{\sin}}$ an unbounded operator $\mathscr{D}_{\boldsymbol{A,}\Phi,Q}$ in $L^{2} (\mathbb{R}^{2},\mathbb{C}^{2})$ generated by $\mathfrak{D}_{\boldsymbol{a} ,\Phi}$ with a domain in $H^{1}(\Omega_{+},\mathbb{C}^{2})\oplus H^{1} (\Omega_{-},\mathbb{C}^{2})$ consisting of functions satisfying interaction conditions on $\Gamma.$ We study the self-adjointness of the operator $\mathscr{D}_{\boldsymbol{A,}\Phi,Q}$ and its essential spectrum for potentials and curves $\Gamma$ slowly oscillating at infinity. We also study the splitting of the interaction problems into two boundary problems describing the confinement of particles in the domains $\Omega_{\pm}.$