Polynomially Superintegrable Hamiltonians Separating in Cartesian Coordinates
Ian Marquette, Anthony Parr
TL;DR
The paper develops a unified, algorithmic approach to construct higher-order superintegrable Hamiltonians that separate in Cartesian coordinates, eliminating the need to classify potentials a priori as standard or exotic. By treating the integral’s homogeneous momentum components and iteratively solving a hierarchy of equations, the authors express the invariant as a rational function in the momenta and derive detailed compatibility conditions that constrain the potential summands $V_i$. In two dimensions, the approach yields explicit linear, quadratic, and cubic compatibility equations, enabling the complete determination of fourth-order standard potentials and several new models, including rational ones. The framework simultaneously encompasses classical and quantum regimes through a deformation-quantization perspective, with results supported by explicit constructions and a Mathematica-based workflow. Overall, the work provides a constructive pipeline for enumerating Cartesian-separable superintegrable systems and their symmetry algebras, while outlining the challenges of extending to higher orders and other coordinate systems.
Abstract
The problem of finding superintegrable Hamiltonians and their integrals of motion can be reduced to solving a series of compatibility equations that result from the overdetermination of the commutator or Poisson bracket relations. The computation of the compatibility equations requires a general formula for the coefficients, which in turn must depend on the potential to be solved for. This is in general a nonlinear problem and quite difficult. Thus, research has focused on dividing the classes of potential into standard and exotic ones so that a number of parameters may be set to zero and the coefficients may be obtained in a simpler setting. We have developed a new method in both the classical and quantum setting which readily yields a formula for the coefficients of the invariant without recourse to this division in the case of Cartesian-separable Hamiltonians. Even though they allow separation of variables as they in general involve potential in terms of higher transcendental and beyond hypergeometric for their wavefunctions, they are quite non-trivial models. The expressions we obtain are in general non-polynomial in the momenta whose fractional terms can be arbitrarily set to zero. These conditions are equivalent to the compatibility equations, but the only unknowns in addition to the potential are constant parameters. We also give the fourth-order standard potentials, and conjectures about general families.