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

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 $sl(2)\oplus sl(2)$.

💡 Research Summary

The paper investigates a broad class of quantum integrable systems whose potentials are trigonometric functions invariant under affine Weyl groups. By passing from the original Cartesian coordinates to the space of invariants (the “orbit space”), the authors obtain a remarkably simple description: the Laplace‑Beltrami operator becomes a second‑order differential operator with a rational metric, and the potential itself is a rational function of the exponential invariants (or polynomial invariants in the rational case). This reformulation reveals a universal algebraic structure underlying all such models.

Two families are distinguished. For the classical families (A‑B‑C‑D), the gauge‑rotated Hamiltonian can be written in terms of the universal enveloping algebra (U(gl_n)). The hidden algebra is generated by a finite set of first‑ and second‑order differential operators that close under commutation. Consequently the spectrum is exactly solvable: eigenfunctions are multivariate polynomials in the invariant variables, and eigenvalues are quadratic polynomials in the quantum numbers. The factorisation property of the eigenfunctions follows directly from the representation theory of (gl_n).

For the exceptional families (G‑F‑E), the situation is richer. The authors construct new infinite‑dimensional, finitely‑generated algebras (\mathcal{A}) of differential operators that replace (U(gl_n)). These algebras contain first‑, second‑ and third‑order generators whose commutation relations are non‑linear but close within (\mathcal{A}). The presence of (\mathcal{A}) explains the appearance of quasi‑exactly solvable (QES) sectors: only a finite part of the spectrum can be obtained algebraically, while the rest remains inaccessible by the same method. The eigenfunctions in the QES sector factorise into products of lower‑degree polynomials, reflecting the underlying (\mathcal{A}) representation.

A particularly illuminating example is the one‑dimensional (BC_1) model, which possesses (\mathbb{Z}_2\oplus T) symmetry. By introducing the invariant variable (z=\cos 2\theta), the authors show that the celebrated TTW (Trigonometric‑Trigonometric‑Wang) model is nothing but the (BC_1) system expressed in orbit coordinates. This insight leads to a new planar QES model whose hidden symmetry algebra is (sl(2)\oplus sl(2)). In this model the wavefunctions separate into a product of two (sl(2))‑type polynomial solutions, and the Hamiltonian can be written as a sum of two commuting (sl(2)) Casimir operators.

Overall, the paper achieves a unifying description of affine‑Weyl‑invariant trigonometric quantum models. It demonstrates that, after gauge rotation and passage to the orbit space, every model admits an algebraic form of the Hamiltonian, a rational potential, and a hidden algebraic structure—(U(gl_n)) for the classical series and novel infinite‑dimensional algebras for the exceptional series. The work not only clarifies the algebraic origin of exact solvability and QES behaviour but also provides concrete constructions (e.g., the (BC_1)‑derived TTW model) that enrich the landscape of integrable quantum systems.