标题： 广义Liénard系统和等时连接 显示英文标题

标题： Generalized Liénard systems and isochronous connections

摘要： 在本文中，我们探讨了非线性Liénard方程$\ddot{x} + k x \dot{x} + \omega^2 x + (k^2/9) x^3 = 0$的经典和量子方面，其中$x=x(t)$是一个实变量，$k, \omega \in \mathbb{R}$。我们证明该方程可以从Levinson-Smith类型的方程导出，其形式为$\ddot{z} + J(z) \dot{z}^2 + F(z) \dot{z} + G(z) = 0$，其中$z=z(t)$是一个实变量，$\{J(z), F(z), G(z)\}$是需要指定的适当函数。通过使用非局部变换，可以进一步将其映射到简谐振子，从而确立其等时性。利用Jacobi最后乘数进行计算表明，该系统表现出双哈密顿特性，即有两个不同类型的哈密顿量描述该系统。对于每一个哈密顿量，我们在动量表象中进行规范量子化，并探索束缚态的可能性。虽然其中一个哈密顿量显示出等间距谱并具有无限状态塔，另一个哈密顿量则表现出分支，但对于某些参数选择可以以闭合形式精确求解。 显示英文摘要

摘要： In this paper, we explore some classical and quantum aspects of the nonlinear Li\'enard equation $\ddot{x} + k x \dot{x} + \omega^2 x + (k^2/9) x^3 = 0$, where $x=x(t)$ is a real variable and $k, \omega \in \mathbb{R}$. We demonstrate that such an equation could be derived from an equation of the Levinson-Smith kind which is of the form $\ddot{z} + J(z) \dot{z}^2 + F(z) \dot{z} + G(z) = 0$, where $z=z(t)$ is a real variable and $\{J(z), F(z), G(z)\}$ are suitable functions to be specified. It can further be mapped to the harmonic oscillator by making use of a nonlocal transformation, establishing its isochronicity. Computations employing the Jacobi last multiplier reveal that the system exhibits a bi-Hamiltonian character, i.e., there are two distinct types of Hamiltonians describing the system. For each of these, we perform a canonical quantization in the momentum representation and explore the possibility of bound states. While one of the Hamiltonians is seen to exhibit an equispaced spectrum with an infinite tower of states, the other one exhibits branching but can be solved exactly in closed form for certain choices of the parameters.