Fredholm property and essential spectrum of 3-D Dirac operators with regular and singular potentials

ABSTRACT

We consider the 3-D Dirac operator with variable regular magnetic and electrostatic potentials, and singular potentials (1) ${\mathfrak{D}}_{\mathit{A},\mathrm{\Phi },Q_{\sin }}\mathit{u}\left(x\right)=\left({\mathfrak{D}}_{\mathit{A},\mathrm{\Phi }}+Q_{\sin }\right)\mathit{u}\left(x\right),x\in {\mathbb{R}}^3$(1) where (2) ${\mathfrak{D}}_{\mathit{A},\mathrm{\Phi }}=\sum _{j=1}^3{\alpha }_j\left(i{\mathrm{\partial }}_{x_j}+A_j\left(x\right)\right)+{\alpha }_{0}m+\mathrm{\Phi }\left(x\right)I_4,$(2) $Q_{\sin }=\mathrm{\Gamma }\left(s\right){\delta }_{\mathrm{\Sigma }}$ is the singular potential with $\mathrm{\Gamma }\left(s\right)={\left({\mathrm{\Gamma }}_{ij}\left(s\right)\right)}_{i,j=1}^4$ being a $4\times 4$ matrix and ${\delta }_{\mathrm{\Sigma }}$ is the delta-function with support on a surface $\mathrm{\Sigma }\subset {\mathbb{R}}^3$ which divides ${\mathbb{R}}^3$ on two open domains ${\mathrm{\Omega }}_{\pm }$ with the common boundary Σ, $\mathit{u}$ is a vector-function on ${\mathbb{R}}^3$ with values in ${\mathbb{C}}^4,{\alpha }_j,j=0,1,2,3$ are the standard $4\times 4$ Dirac matrices. We associate with the formal Dirac operator ${\mathfrak{D}}_{\mathit{A},\mathrm{\Phi },Q_{\sin }}$ an unbounded operator $\mathcal{D}$ in $L^2({\mathbb{R}}^3,{\mathbb{C}}^4)$ generated by ${\mathfrak{D}}_{\mathit{A},\mathrm{\Phi }}$ with domain in $H^1({\mathrm{\Omega }}_{+},{\mathbb{C}}^4)\oplus H^1({\mathrm{\Omega }}_{-},{\mathbb{C}}^4)$ consisting of functions satisfying transmission conditions on Σ. We consider the self-adjointness of operator $\mathcal{D}$, its Fredholm properties, and the essential spectrum in the case if Σ is either a closed $C^2$-surface or an unbounded $C^2$-hypersurface with a regular behaviour at infinity.

As application we consider the electrostatic and Lorentz scalar δ-shell interactions.

Keywords:

• Dirac operators
• singular potentials
• self-adjointness
• essential spectrum

AMS Subject Classifications:

• 35J10
• 47A10
• 47A53
• 81Q10

Disclosure statement

No potential conflict of interest was reported by the author(s).