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

TL;DR

This paper investigates the spectral properties of a relativistic Landau operator with delta and delta prime interactions on a cylindrical surface, using self-adjoint extension theory to understand the effects of these singular interactions.

Contribution

It introduces a rigorous analysis of the relativistic Landau operator with delta-type interactions on a cylinder using self-adjoint extension methods, which is a novel approach in this context.

Findings

Spectral analysis of the operator with delta interactions

Spectral analysis of the operator with delta prime interactions

Characterization of the spectrum depending on interaction type

Abstract

Using the theory of self-adjoint extensions, we study the relativistic spectral properties of the Landau operator with $\delta$ and $\delta '$ interactions on a cylinder of radius $R$ for a charged spin particle system, formally given by the Hamiltonian $H^G_{B} = {(p-A)}^2 I + {\bf{\sigma}}.{\bf{B}+G}V(r),$ $\bigg(V(r) = \delta (R-r)$ or $V(r) = \delta' (R-r) \bigg),$ 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 $B>0.$