Quasi-exact-solvability of the $A_{2}/G_2$ Elliptic model: algebraic forms, $sl(3)/g^{(2)}$ hidden algebra, polynomial eigenfunctions

TL;DR

This paper demonstrates the quasi-exact solvability of the $A_2$ and $G_2$ elliptic models by revealing their hidden algebraic structures, polynomial eigenfunctions, and explicit algebraic forms through variable transformations and algebraic analysis.

Contribution

It introduces algebraic forms and hidden $sl(3)$ and $g^{(2)}$ algebras for the elliptic models, enabling explicit construction of polynomial eigenfunctions and identifying special coupling constants.

Findings

Existence of finite polynomial eigenfunctions at specific coupling constants.

Explicit algebraic forms of the potentials as rational functions.

Identification of hidden $sl(3)$ and $g^{(2)}$ algebraic structures.

Abstract

The potential of the $A_2$ quantum elliptic model (3-body Calogero-Moser elliptic model) is defined by the pairwise three-body interaction through Weierstrass $\wp$-function and has a single coupling constant. A change of variables has been found, which are $A_2$ elliptic invariants, such that the potential becomes a rational function, while the flat space metric as well as its associated vector are polynomials in two variables. It is shown that the model possesses the hidden $sl(3)$ algebra - the Hamiltonian is an element of the universal enveloping algebra $U_{sl(3)}$ for arbitrary coupling constant - thus, it is equivalent to $sl(3)$-quantum Euler-Arnold top. The integral, in a form of the third order differential operator with polynomial, is constructed explicitly, being also an element of $U_{sl(3)}$. It is shown that there exists a discrete sequence of the coupling constants for which a finite number of polynomial eigenfunctions, up to a (non-singular) gauge factor occur. The potential of the $G_2$ quantum elliptic model (3-body Wolfes elliptic model) is defined by the pairwise and three-body interactions through Weierstrass $\wp$-function and has two coupling constants. A change of variables has been found, which are $G_2$ elliptic invariants, such that the potential becomes a rational function, while the flat space metric as well as its associated vector are polynomials in two variables. It is shown the model possesses the hidden $g^{(2)}$ algebra. It is shown that there exists a discrete family of the coupling constants for which a finite number of polynomial eigenfunctions up to a (non-singular) gauge factor occur.