Fifth-order superintergrable quantum system separating in Cartesian coordinates. Doubly exotic potentials

Fourth order Superintegrable systems separating in Cartesian coordinates I. Exotic quantum potentials

A study is presented of two-dimensional superintegrable systems separating in Cartesian coordinates and allowing an integral of motion that is a fourth order polynomial in the momenta. All quantum mechanical potentials that do not satisfy any linear differential equation are found. They do however satisfy nonlinear ODEs. We show that these equations always have the Painlevé property and integrate them in terms of known Painlevé transcendents or elliptic functions.

Doubly Exotic $N$th-order Order Superintegrable Classical Systems Separating in Cartesian Coordinates

Superintegrable classical Hamiltonian systems in two-dimensional Euclidean space $E_2$ are explored. The study is restricted to Hamiltonians allowing separation of variables $V(x,y)=V_1(x)+V_2(y)$ in Cartesian coordinates. In particular, the Hamiltonian $\mathcal H$ admits a polynomial integral of order $N>2$. Only doubly exotic potentials are considered. These are potentials where none of their separated parts obey any linear ordinary differential equation. An improved procedure to calculate these higher-order superintegrable systems is described in detail. The two basic building blocks of th…

New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates

A study is presented of superintegrable quantum systems in two-dimensional Euclidean space $E_2$ allowing the separation of variables in Cartesian coordinates. In addition to the Hamiltonian $H$ and the second order integral of motion $X$, responsible for the separation of variables, they allow a third integral that is a polynomial of order $N\, (N\geq3)$ in the components $p_1, p_2$ of the linear momentum. We focus on doubly exotic potentials, i.e. potentials $V(x, y) = V_1(x) + V_2(y)$ where neither $V_1(x)$ nor $V_2(y)$ satisfy any linear ordinary differential equation. We present two new i…

基于高阶阶梯算符的笛卡尔坐标可分离超可积系统的多项式代数

Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

该研究引入了表征笛卡尔坐标可分离高阶超可积系统的通用多项式代数，基于多项式Heisenberg代数与高阶阶梯算符构建，保留gl(n)李代数部分结构，并以新型三维超可积系统为例，通过代数的有限维不可约表示确定了模型简并度。

多项式超可积哈密顿量在笛卡尔坐标中的分离

Polynomially Superintegrable Hamiltonians Separating in Cartesian Coordinates

本文提出了一种新方法，用于计算笛卡尔坐标中分离变量的超可积哈密顿量的系数，无需将势能分为标准和非标准类别，并给出了四阶标准势能及一般族的猜想。

Third-order superintegrable systems with potentials satisfying nonlinear equations

The conditions for superintegrable systems in two-dimensional Euclidean space admitting separation of variables in an orthogonal coordinate system and a functionally independent third-order integral are studied. It is shown that only systems that separate in subgroup type coordinates, Cartesian or polar, admit potentials that can be described in terms of nonlinear special functions. Systems separating in parabolic or elliptic coordinated are shown to have potentials with only non-movable singularities.