标题： 具有奇异位置依赖质量的等时振子及其量子化 显示英文标题

标题： Isochronous oscillator with a singular position-dependent mass and its quantization

摘要： 在本文中，我们对方程$\ddot{x} - (1/2x) \dot{x}^2 + 2 \omega^2 x - 1/8x = 0$进行分析，其中$\omega > 0$和$x = x(t)$是一个实值变量。我们首先讨论该方程从一种与位置有关的质量情形中出现的情况，在这种情况下，质量分布与$x$成反比，在$x = 0$处允许存在奇点。 相关的势能也在$x = 0$处是奇异的，将实轴分成两部分，即$x > 0$和$x < 0$。 对于两个分支的动力学都是精确可解的，因此为了明确起见，我们专注于$x > 0$分支。 在位置表象中进行规范量子化，并采用 von Roos 提出的动能算符的排序策略，我们证明该问题与等频振子是等谱的。 因此，量子谱由无限多个等间距的能级组成。 能级之间的间距被发现对用于按 von Roos 方式对动能算符进行排序的模糊参数的具体选择不敏感。 显示英文摘要

摘要： In this paper, we present an analysis of the equation $\ddot{x} - (1/2x) \dot{x}^2 + 2 \omega^2 x - 1/8x = 0$, where $\omega > 0$ and $x = x(t)$ is a real-valued variable. We first discuss the appearance of this equation from a position-dependent-mass scenario in which the mass profile goes inversely with $x$, admitting a singularity at $x = 0$. The associated potential is also singular at $x = 0$, splitting the real axis into two halves, i.e., $x > 0$ and $x < 0$. The dynamics is exactly solvable for both the branches and so for definiteness, we stick to the $x > 0$ branch. Performing a canonical quantization in the position representation and upon employing the ordering strategy of the kinetic-energy operator due to von Roos, we show that the problem is isospectral to the isotonic oscillator. Thus, the quantum spectrum consists of an infinite number of equispaced levels. The spacing between the energy levels is found to be insensitive to the specific choices of the ambiguity parameters that are employed for ordering the kinetic-energy operator \`a la von Roos.