Comments on dihedral and supersymmetric extensions of a family of Hamiltonians on a plane

TL;DR

This paper explores the relationship between dihedral and supersymmetric extensions of specific Hamiltonians on a plane, using group theory and fermionic operators to establish connections and identities.

Contribution

It introduces a novel connection between dihedral and supersymmetric extensions of Hamiltonians using fermionic operators and trigonometric identities.

Findings

Established a link between dihedral and supersymmetric Hamiltonian extensions

Realized dihedral group elements via fermionic creation and annihilation operators

Derived new trigonometric identities related to the Hamiltonians

Abstract

For any odd $k$, a connection is established between the dihedral and supersymmetric extensions of the Tremblay-Turbiner-Winternitz Hamiltonians $H_k$ on a plane. For this purpose, the elements of the dihedral group $D_{2k}$ are realized in terms of two independent pairs of fermionic creation and annihilation operators and some interesting trigonometric identities are demonstrated.