Boundary Problems for Three-Dimensional Dirac Operators and Generalized MIT Bag Models for Unbounded Domains

Abstract

We consider the operators of the following boundary problems

$$\mathbb{D}_{\boldsymbol{A,}\Phi,\mathfrak{B}}\boldsymbol{u}=\left\{ \begin{array} [c]{c} \mathfrak{D}_{\boldsymbol{A},\Phi}\boldsymbol{u}\text{ on }\Omega\\ \mathfrak{B}\boldsymbol{u}_{\partial\Omega}\text{ on }\partial\Omega \end{array} \right. $$

(1)

in unbounded domains $\Omega\subset\mathbb{R}^{3}$, where $\mathfrak{D} _{\boldsymbol{A},\Phi}$ is the $3-D$ Dirac operator

$$\mathfrak{D}_{\boldsymbol{A},\Phi}=\left[ \boldsymbol{\alpha\cdot }(i\boldsymbol{\nabla}+\boldsymbol{A})+\alpha_{0}m+\Phi I_{4}\right]=\sum_{j=1}^{3}\alpha_{j}(i\partial_{x_{j}}+A_{j})\alpha_{0}m+\Phi I_{4}$$

defined on the distributions $\boldsymbol{u}=(u_{1},u_{2},u_{3},u_{4})\in H^{1}(\Omega,\mathbb{C}^{4})$, where $\alpha_{0},\alpha_{1},\alpha_{2} ,\alpha_{3}$ are Dirac matrices, $\boldsymbol{A}\in L^{\infty} (\Omega,\mathbb{R}^{3})$ and $\Phi\in L^{\infty}(\mathbb{R}^{3})$ are the magnetic and electrostatic potentials, $m\in\mathbb{R}$ is the mass of a particle. Let $\mathbb{C}^{4}\ni\boldsymbol{u}=(\boldsymbol{u}^{1} ,\boldsymbol{u}^{2})\in\mathbb{C}^{2}\oplus\mathbb{C}^{2}.$ We assume that the operator $\mathfrak{B}$ of the boundary condition is

$$\mathfrak{B}\boldsymbol{u}_{\partial\Omega}=\mathfrak{b}_{1}\boldsymbol{u} _{\partial\Omega}^{1}+\mathfrak{b}_{2}\boldsymbol{u}_{\partial\Omega}^{2}, $$

(2)

where $\mathfrak{b}_{j},j=1,2$, are $2\times2$ matrices, $\boldsymbol{u} _{\partial\Omega}^{j}\in H^{1/2}(\partial\Omega,\mathbb{C}^{2}),j=1,2$, are restrictions of distributions $\boldsymbol{u}^{j}\in H^{1}(\Omega ,\mathbb{C}^{2})$ on $\partial\Omega.$ The class of the boundary condition (2) in a particular case contains the boundary conditions of the MIT bag model and its generalizations which describe the confinement of the quarks to the domain $\Omega.$

We give conditions of self-adjointness of unbounded operators $\mathscr{D} _{\boldsymbol{A,}\Phi,\mathfrak{B}}$ associated with the boundary problem (1) and give a description of the essential spectrum of $\mathscr{D} _{\boldsymbol{A,}\Phi,\mathfrak{B}}$ for certain unbounded domains by applying the limit operators method.