Boundary Value Problems for 3D-Dirac Operators and MIT Bag Model

Abstract

We consider the operator $\mathbb {D}_{\boldsymbol {A,}\Phi ,\mathfrak {B}}$ of boundary value problem

$$\displaystyle \mathbb {D}_{\boldsymbol {A,}\Phi ,\mathfrak {B}}\boldsymbol {u=} \left \{{\begin {array}[c]{c} \mathfrak {D}_{\boldsymbol {A},\Phi }\boldsymbol {u}\mbox{ on }\Omega \\ \mathfrak {B}\boldsymbol {u=u}_{\partial \Omega }^{(2)}-\mathfrak {b} \boldsymbol {u}_{\partial \Omega }^{(1)}\ \mbox{on }\partial \Omega \end {array}}\right . $$

(1)

for 3D −Dirac operators $\mathfrak {D}_{\boldsymbol {A},\Phi }$ with the magnetic potential $\boldsymbol {A}\in L^{\infty }(\Omega ,\mathbb {R}^{3}),$ and electric potential Φ ∈ L ^∞( Ω) in domain $\Omega \subset \mathbb {R}^{3}$ with C ²-uniformly regular boundary, where $\boldsymbol {u=}\left ( \begin {array} [c]{c} \boldsymbol {u}^{(1)}\\ \boldsymbol {u}^{(2)} \end {array} \right ) \in \mathbb {C}^{4}$ be a vector with $\boldsymbol {u}^{(j)} \in \mathbb {C}^{2},j=1,2,\boldsymbol {u}_{\partial \Omega }^{(j)}$ is the boundary value of u ^(j) on ∂ Ω, where $\mathfrak {b}$ is a 2 × 2 matrix with entries $\mathfrak {b}_{ij}$ belonging to the space of bounded continuous functions on ∂ Ω. We associate with this boundary value problem an unbounded operator $\mathcal {D}_{\boldsymbol {A} ,\Phi ,\mathfrak {B}}$ in $L^{2}(\Omega ,\mathbb {C}^{4})$ defined by the Dirac operator $\mathfrak {D}_{\boldsymbol {A},\Phi }$ on Ω with domain

$$\displaystyle dom\mathcal {D}_{\boldsymbol {A},\Phi ,\mathfrak {B}}=\left \{ \boldsymbol {u}\in H^{1}(\Omega ,\mathbb {C}^{4}):\boldsymbol {u}_{\partial \Omega }^{(2)} -\mathfrak {b}\boldsymbol {u}_{\partial \Omega }^{(1)}=\boldsymbol {0}\mbox{ on }\partial \Omega \right \} . $$

We obtain conditions of the self-adjointness of $\mathcal {D}_{\boldsymbol {A} ,\Phi ,\mathfrak {B}}$ and the discreteness of its spectrum. We give applications of this results to the operator of MIT bag model of relativistic quantum mechanics.