Exchange operator formalism for an infinite family of solvable and integrable quantum systems on a plane

TL;DR

This paper extends the exchange operator formalism to an infinite family of exactly solvable and integrable quantum systems on a plane, utilizing dihedral group symmetries to construct and analyze these Hamiltonians.

Contribution

It generalizes the exchange operator formalism to an infinite set of Hamiltonians using dihedral group symmetries, enabling the construction of new solvable quantum models.

Findings

Developed differential-difference operators $D_r$ and $D_{}$ for $D_{2k}$ groups.

Constructed $D_{2k}$-extended Hamiltonians $ ilde{H}_k$ with invariance properties.

Retrieved original Hamiltonians $H_k$ via projection in the $D_{2k}$ representation space.

Abstract

The exchange operator formalism in polar coordinates, previously considered for the Calogero-Marchioro-Wolfes problem, is generalized to a recently introduced, infinite family of exactly solvable and integrable Hamiltonians $H_k$, $k=1$, 2, 3,..., on a plane. The elements of the dihedral group $D_{2k}$ are realized as operators on this plane and used to define some differential-difference operators $D_r$ and $D_{\varphi}$. The latter serve to construct $D_{2k}$-extended and invariant Hamiltonians $\chh_k$, from which the starting Hamiltonians $H_k$ can be retrieved by projection in the $D_{2k}$ identity representation space.