标题： 具有奇异位置相关质量的等时振荡器及其量子化 显示英文标题

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

摘要： 本文中，我们分析了方程$\ddot{x} - (1/2x) \dot{x}^2 + 2 \omega^2 x - 1/8x = 0$，其中$\omega > 0$和$x = x(t)$是实值变量。 我们首先讨论了该方程出现在一个位置相关质量的场景中，此时质量分布与$x$成反比，并在$x = 0$处出现奇点的情况。 与之相关的势也在$x = 0$处发散，将实轴分成两部分，即$x > 0$和$x < 0$。 这两个分支的动力学问题都可以精确求解，因此为了明确起见，我们专注于$x > 0$分支。 在位置表象下进行规范量子化，并采用 von Roos 提出的动能算符的排序策略后，我们证明了该问题与各向同性振子是等谱的。 因此，量子能谱由无穷多个等间距的能级组成，这符合经典系统支持等时振荡时，在其量子版本中对应的能级也是等间距的预期。 研究发现，能级之间的间距对 von Roos 提出的用于排列动能算符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 with the isotonic oscillator. Thus, the quantum spectrum consists of an infinite number of equispaced levels which is in line with the expectation that classical systems that support isochronous oscillations are associated in their quantized versions with equally-spaced spectra. The spacing between 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.