Haantjes algebras, Zernike system and separation of variables
Abstract
The generalized Zernike family H(N) = p12 + p22 + ∑n=1N γn (q1 p1 + q2 p2)n is a parametric family of two-dimensional superintegrable Hamiltonians, admitting N integrals of motion of degree N in the momenta. A theorem of Nozaleda, Tempesta, and Tondo guarantees that canonical separation coordinates (Darboux--Haantjes coordinates) exist for any such system; the challenge is to construct them explicitly. This paper solves the problem for N = 2 -- the classical Zernike system, which is canonically equivalent to the isotropic harmonic oscillator on flat space or on a space of constant curvature -- covering all four known separation types: polar, two Cartesian-type, and elliptic. The key structural fact is that the Haantjes operators associated with all integrals of H(2) have no momentum-dependent off-diagonal block (lift form). We prove that this implies the separation coordinates are reachable by an extended point transformation: the new positions depend only on the old positions, with no momentum entering the coordinate change. In the polar and Cartesian-type cases the new position coordinates involve at most a square root of a single-variable rational function; in the elliptic case they are given by the two roots of a quadratic polynomial in the original coordinates, and the resulting branch structure introduces a fourth regular singular point in the quantum separated ODE, placing it in the Heun class, in agreement with results of Atakishiyev, Pogosyan, Vicent, Wolf, and Yakhno. For N ≥ 3 we prove an obstruction: no lift-form Haantjes operator can generate an integral independent of the angular momentum. The separation coordinates for higher Zernike Hamiltonians therefore require momentum-dependent canonical transformations, whose explicit construction is the subject of future work.
Versions
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BibTeX
@misc{kubu2026haantjes,
title = {{Haantjes algebras, Zernike system and separation of variables}},
author = {Ondřej Kubů and Danilo Latini},
howpublished = {arXiv},
year = {2026},
month = may,
doi = {10.48550/arxiv.2605.23748},
eprint = {2605.23748},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.23748},
}
RIS
TY - UNPB
TI - Haantjes algebras, Zernike system and separation of variables
AU - Ondřej Kubů
AU - Danilo Latini
PY - 2026
DA - 2026/05/25
PB - arXiv
DO - 10.48550/arxiv.2605.23748
UR - https://arxiv.org/abs/2605.23748
AB - The generalized Zernike family H_(N) = p_1^2 + p_2^2 + ∑_n=1^N γ_n (q_1 p_1 + q_2 p_2)^n is a parametric family of two-dimensional superintegrable Hamiltonians, admitting N integrals of motion of degree N in the momenta. A theorem of Nozaleda, Tempesta, and Tondo guarantees that canonical separation coordinates (Darboux--Haantjes coordinates) exist for any such system; the challenge is to construct them explicitly. This paper solves the problem for N = 2 -- the classical Zernike system, which is canonically equivalent to the isotropic harmonic oscillator on flat space or on a space of constant curvature -- covering all four known separation types: polar, two Cartesian-type, and elliptic. The key structural fact is that the Haantjes operators associated with all integrals of H_(2) have no momentum-dependent off-diagonal block (lift form). We prove that this implies the separation coordinates are reachable by an extended point transformation: the new positions depend only on the old positions, with no momentum entering the coordinate change. In the polar and Cartesian-type cases the new position coordinates involve at most a square root of a single-variable rational function; in the elliptic case they are given by the two roots of a quadratic polynomial in the original coordinates, and the resulting branch structure introduces a fourth regular singular point in the quantum separated ODE, placing it in the Heun class, in agreement with results of Atakishiyev, Pogosyan, Vicent, Wolf, and Yakhno. For N ≥ 3 we prove an obstruction: no lift-form Haantjes operator can generate an integral independent of the angular momentum. The separation coordinates for higher Zernike Hamiltonians therefore require momentum-dependent canonical transformations, whose explicit construction is the subject of future work.
ER -