Dirac electron in graphene with magnetic fields arising from first-order intertwining operators

Abstract

The behaviour of a Dirac electron in graphene, under magnetic fields which are orthogonal to the layer, is studied. The initial problem is reduced to an equivalent one, where two one-dimensional Schrödinger Hamiltonians $H^{\pm }$ are intertwined by a first order differential operator. Special magnetic field are initially chosen, in order that $V^{\pm }$ will be shape invariant exactly solvable potentials. When looking for more general first order operators, intertwining $H^{-}$ with a non-necessarily shape invariant Hamiltonian, new magnetic fields associated also to analytic solutions will be generated. The iteration of this procedure is as well discussed.

Several misprints corrected