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.

中文翻译：

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三维Dirac算子的边界问题和无界域的广义MIT袋模型

摘要

我们考虑以下边界问题的算子

$$ \ mathbb {D} _ {\ boldsymbol {A，} \ Phi，\ mathfrak {B}} \ boldsymbol {u} = \ left \ {\ begin {array} [c] {c} \ mathfrak {D} _ {\ boldsymbol {A}，\ Phi} \ boldsymbol {u} \ text {on} \ Omega \\ \ mathfrak {B} \ boldsymbol {u} _ {\ partial \ Omega} \ text {on} \ partial \欧米茄\ end {array} \ right。$$（1）

在无界域\（\ Omega \ subset \ mathbb {R} ^ {3} \）中，其中\（\ mathfrak {D} _ {\ boldsymbol {A}，\ Phi} \）是\（3-D \ ） Dirac运算符

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

定义在分布\（\ boldsymbol {u} = {u_ {1}，u_ {2}，u_ {3}，u_ {4}）\中H ^ {1}（\ Omega，\ mathbb {C} ^ {4}）\），其中\（\ alpha_ {0}，\ alpha_ {1}，\ alpha_ {2}，\ alpha_ {3} \）是Dirac矩阵，\（\ boldsymbol {A} \ in L ^ {\ infty}（\ Omega，\ mathbb {R} ^ {3}）\）和\（\ Phi \ in L ^ {\ infty}（\ mathbb {R} ^ {3}）\）具有磁性，静电势\（m \ in \ mathbb {R} \）是粒子的质量。让\（\ mathbb {C} ^ {4} \ ni \ boldsymbol {u} =（\\ boldsymbol {u} ^ {1}，\ boldsymbol {u} ^ {2}）\ in \ mathbb {C} ^ { 2} \ oplus \ mathbb {C} ^ {2}。\）我们假设边界条件的运算符\（\ mathfrak {B} \）为

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

其中\（\ mathfrak {b} _ {j}，j = 1,2 \）是\（2 \ times2 \）矩阵，\（\ boldsymbol {u} _ {\ partial \ Omega} ^ {j} \在H ^ {1/2}（\ partial \ Omega，\ mathbb {C} ^ {2}），j = 1,2 \）中，是分布的限制\（\ boldsymbol {u} ^ {j} \ in \（\ partial \ Omega。\）上的H ^ {1}（\ Omega，\ mathbb {C} ^ {2}）\）在特定情况下，边界条件（2）的类别包含边界条件的边界条件。 MIT袋模型及其泛化描述了夸克在域\（\ Omega。\）中的限制

我们给出与边界问题（1）关联的无界算子\（\ mathscr {D} _ {\ boldsymbol {A，} \ Phi，\ mathfrak {B}} \}的自伴随条件，并给出关于通过应用极限算子方法，对某些无界域，\（\ mathscr {D} _ {\ boldsymbol {A，} \ Phi，\ mathfrak {B}} \}的基本谱。