# New solutions for graphene with scalar potentials by means of generalized intertwining

## TL;DR

This paper develops a generalized intertwining method to analytically solve the two-dimensional Dirac equation with scalar potentials, relevant for modeling electrons in graphene under external electrostatic fields.

## Contribution

It introduces a novel generalized intertwining approach to find exact solutions for the Dirac equation with scalar potentials in graphene.

## Key findings

- Analytical solutions for wave functions in scalar potential fields.
- Explicit examples demonstrating the method's application.
- Potential for modeling electronic properties in graphene.

## Abstract

The intertwining relations between superpartner Hamiltonians are the main ingredients of well known Supersymmetrical Quantum Mechanics (SUSY QM). In the present paper, the generalized form of intertwining is used for investigation of a massless (zero energy) two-dimensional Dirac equation with scalar external potential. This equation is related to the description of graphene and some other materials in the field of external electrostatic potential. The use of modified intertwining relations allows to find analytically solutions for the wave functions in the field of some external scalar potentials which depend on both space coordinates. A few examples of this construction are given explicitly.