Exactly solvable potentials of Calogero type for q-deformed Coxeter groups

TL;DR

This paper introduces a method to construct exactly solvable quantum models of Calogero type using polynomial invariants of q-deformed Coxeter groups, exemplified by the G2^q case.

Contribution

It demonstrates how to parameterize quantum systems with q-deformed Coxeter invariants to achieve exact solvability, extending previous approaches to new algebraic structures.

Findings

Constructed exactly solvable Calogero-type potentials for q-deformed Coxeter groups.

Explicitly derived the G2^q potential using gauge transformations.

Established a framework linking polynomial invariants to quantum solvability.

Abstract

We establish that by parameterizing the configuration space of a one-dimensional quantum system by polynomial invariants of q-deformed Coxeter groups it is possible to construct exactly solvable models of Calogero type. We adopt the previously introduced notion of solvability which consists of relating the Hamiltonian to finite dimensional representation spaces of a Lie algebra. We present explicitly the $G_2^q $-case for which we construct the potentials by means of suitable gauge transformations.