Two-Dimensional Dirac Operators with Interactions on Unbounded Smooth Curves
Vladimir Rabinovich
Abstract
We consider the 2D Dirac operator with singular potentials 1 $$\mathfrak{D}_{\boldsymbol{A},\Phi,Q_{\sin}}\boldsymbol{u}(x)=\left( \mathfrak{D}_{\boldsymbol{A},\Phi}+Q_{\sin}\right) \boldsymbol{u} (x),\quad x\in\mathbb{R}^{2}, $$ where 2 $$\mathfrak{D}_{\boldsymbol{a},\Phi}= {\displaystyle\sum\limits_{j=1}^{2}} \sigma_{j}\left( i\partial_{x_{j}}+a_{j}\right) +\sigma_{3}m+\Phi I_{2}; $$ 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 $$\Omega_{\pm}.$$