From Quantum $A_N$ to $E_8$ Trigonometric Model: Space-of-Orbits View

TL;DR

This paper explores affine-Weyl-invariant quantum models with trigonometric potentials, revealing their algebraic structures, solvability properties, and connections to known models like TTW, advancing understanding of integrable systems in mathematical physics.

Contribution

It introduces a unified algebraic framework for affine-Weyl-invariant models, including new algebraic structures for exceptional cases and a link to the TTW model.

Findings

Models are completely integrable with polynomial eigenfunctions.

A hidden algebraic structure related to universal enveloping algebras is identified.

New quasi-exactly solvable model with $sl(2) imes sl(2)$ symmetry is constructed.

Abstract

A number of affine-Weyl-invariant integrable and exactly-solvable quantum models with trigonometric potentials is considered in the space of invariants (the space of orbits). These models are completely-integrable and admit extra particular integrals. All of them are characterized by (i) a number of polynomial eigenfunctions and quadratic in quantum numbers eigenvalues for exactly-solvable cases, (ii) a factorization property for eigenfunctions, (iii) a rational form of the potential and the polynomial entries of the metric in the Laplace-Beltrami operator in terms of affine-Weyl (exponential) invariants (the same holds for rational models when polynomial invariants are used instead of exponential ones), they admit (iv) an algebraic form of the gauge-rotated Hamiltonian in the exponential invariants (in the space of orbits) and (v) a hidden algebraic structure. A hidden algebraic structure for $(A-B-C{-D)$-models, both rational and trigonometric, is related to the universal enveloping algebra $U_{gl_n}$. For the exceptional $(G-F-E)$-models, new, infinite-dimensional, finitely-generated algebras of differential operators occur. Special attention is given to the one-dimensional model with $BC_1\equiv(\mathbb{Z}_2)\oplus T$ symmetry. In particular, the $BC_1$ origin of the so-called TTW model is revealed. This has led to a new quasi-exactly solvable model on the plane with the hidden algebra $sl(2)\oplus sl(2)$.