Tremblay-Turbiner-Winternitz (TTW) system at integer index $k$: polynomial algebras of integrals

TL;DR

The paper investigates the TTW quantum system at integer indices k=1,2,3,4, uncovering a hidden g^{(k)} algebra and a 4-generated polynomial algebra of integrals (H,I_1,I_2,I_{12}). For each k, the authors compute the Hamiltonian, the two integrals, and their commutator, showing that the double commutators are finite-degree polynomials in the generators, with a syzygy I_{12}^2 = R_k(H,I_1,I_2,I_{12}). The results demonstrate a consistent algebraic structure across k=1..4, embedded in g^{(k)}, and lead to a conjecture that this hidden algebraic/polynomial framework holds for all integer k, tying TTW to dihedral invariants and exact solvability via polynomial invariant subspaces. The work provides a rigorous algebraic foundation for the superintegrability and exact solvability of TTW and clarifies the role of the nontrivial integral I_2 in the corresponding higher-rank algebras.

Abstract

An infinite 3-parametric family of superintegrable and exactly-solvable quantum models on a plane, admitting separation of variables in polar coordinates, marked by integer index $k$ was introduced in Journ Phys A 42 (2009) 242001 and was called in literature the TTW system. In this paper it is conjectured that the Hamiltonian and both integrals of TTW system have hidden algebra $g^{(k)}$ - it was checked for $k=1,2,3,4$ - having its finite-dimensional representation spaces as the invariant subspaces. It is checked that for $k=1,2,3,4$ that the Hamiltonian $H$, two integrals ${\cal I}_{1,2}$ and their commutator ${\cal I}_{12} = [{\cal I}_1,{\cal I}_2]$ are four generating elements of the polynomial algebra of integrals of the order $(k+1)$: $[{\cal I}_1,{\cal I}_{12}] = P_{k+1}(H, {\cal I}_{1,2},{\cal I}_{12})$, $[{\cal I}_2,{\cal I}_{12}] = Q_{k+1}(H, {\cal I}_{1,2},{\cal I}_{12})$, where $P_{k+1},Q_{k+1}$ are polynomials of degree $(k+1)$ written in terms of ordered monomials of $H, {\cal I}_{1,2},{\cal I}_{12}$. This implies that polynomial algebra of integrals is subalgebra of $g^{(k)}$. It is conjectured that all is true for any integer $k$.