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

## TL;DR

This paper explores the behavior of Dirac electrons in graphene under specific magnetic fields, using intertwining operators to generate exactly solvable potentials and new magnetic field configurations.

## Contribution

It introduces a method to generate new magnetic fields and solutions for Dirac electrons in graphene via first-order intertwining operators, extending beyond shape-invariant potentials.

## Key findings

- Derived new exactly solvable magnetic field configurations.
- Established a framework for generating analytic solutions for Dirac electrons.
- Discussed iterative procedures for constructing complex magnetic fields.

## 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 Schr\"{o}dinger 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.