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

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 ${\mathcal{I}}_{1,2}$ and their commutator ${\mathcal{I}}_{12}=[{\mathcal{I}}_1,{\mathcal{I}}_2]$ are four generating elements of the polynomial algebra of integrals of the order $(k+1)$: $[{\mathcal{I}}_1,{\mathcal{I}}_{12}]=P_{k+1}(H,{\mathcal{I}}_{1,2},{\mathcal{I}}_{12})$, $[{\mathcal{I}}_2,{\mathcal{I}}_{12}]=Q_{k+1}(H,{\mathcal{I}}_{1,2},{\mathcal{I}}_{12})$, where $P_{k+1},Q_{k+1}$ are polynomials of degree $(k+1)$ written in terms of ordered monomials of $H,{\mathcal{I}}_{1,2},{\mathcal{I}}_{12}$. This implies that polynomial algebra of integrals is subalgebra of $g^{(k)}$. It is conjectured that all is true for any integer $k$.

55 pages, 4 Appendices, 24 references, extended version: six typos in formulas found by co-Scientist (Academia.edu) fixed (two in Appendix B)

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