Relativistic Spectral Properties of Landau Operator with $δ$- and $δ'$- cylinder interactions

Abstract

Using the theory of self-adjoint extensions, we study the relativistic spectral properties of the Landau operator with $δ$ and $δ'$ interactions on a cylinder of radius $R$ for a charged spin particle system, formally given by the Hamiltonian $H^G_{B} = {(p-A)}^2 I + {\bfσ}.{\bf{B}+G}V(r),$ $\bigg(V(r) = δ(R-r)$ or $V(r) = δ' (R-r) \bigg),$ acting in $ L^2(\R^2)\bigotimes \C^2 $. $G$ a scalar $2\times 2$ real matrix. I is the identity matrix. The potential vector has the form $A= (B/2)(-y, x)$ and $B>0.$