论文标题
$ n $ th级可促进系统的新无限家庭分隔于笛卡尔坐标
New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates
论文作者
论文摘要
对二维欧几里得空间中的可整合量子系统的研究进行了研究$ e_2 $，允许在笛卡尔坐标中分离变量。除了造成变量分离的Hamiltonian $ h $和MOTIOT $ X $的二阶积分外，它们还允许第三个集成量，这是$ n \，（n \ geq3）$的多项式，（N \ GEQ3）$ p_1，P_2，P_2 linear Momentum的P_2 $。我们专注于双重外来电位，即电势$ v（x，y）= v_1（x） + v_2（y）$，其中$ v_1（x）$也不是$ v_2（y）$满足任何线性的普通微分方程。我们在$ e_2 $中介绍了两个新的无限型系统家族，其中包括$ n $的积分$ n $，其中$ v_1（x）$和$ v_2（y）$由通过Painlevé测试的非线性ODE解决方案提供。以$ 3 \ leq n \ leq 10 $进行验证。我们猜想这将具有任何双重异国情调的潜力和所有$ n $，而且潜力实际上将始终具有Parelevé财产。
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 infinite families of superintegrable systems in $E_2$ with integrals of order $N$ for which $V_1(x)$ and $V_2(y)$ are given by the solution of a nonlinear ODE that passes the Painlevé test. This was verified for $3\leq N \leq 10$. We conjecture that this will hold for any doubly exotic potential and for all $N$, and that moreover the potentials will always actually have the Painlevé property.
