非常曲率空间上超可积势的路径积分方法：I.达布空间DI和DII
Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: I. Darboux Spaces DI and DII
摘要：
本文将Feynman路径积分技巧应用于二维非常曲率空间D_I和D_II上的超可积势。在D_I上有三个，在D_II上有四个.我们能够在大多数分离的坐标系中计算路径积分，从而得到格林函数、离散和连续波函数以及离散能谱的表达式。然而，在某些情况下，离散谱不能被明确地陈述，因为它要么由涉及抛物柱面函数（达布空间I）的超越方程确定，要么由高阶多项式方程确定。特别地，关于DI的解表明超可积系统不一定是退化的。我们还可以展示平坦空间（常曲率为零）和二维双曲面（常负曲率）的极限情况是如何出现的。
英文摘要：
In this paper the Feynman path integral technique is applied for superintegrable potentials on two-dimensional spaces of non-constant curvature: these spaces are Darboux spaces D_I and D_II, respectively. On D_I there are three and on D_II four such potentials, respectively. We are able to evaluate the path integral in most of the separating coordinate systems, leading to expressions for the Green functions, the discrete and continuous wave-functions, and the discrete energy-spectra. In some cases, however, the discrete spectrum cannot be stated explicitly, because it is either determined by a transcendental equation involving parabolic cylinder functions (Darboux space I), or by a higher order polynomial equation. The solutions on D_I in particular show that superintegrable systems are not necessarily degenerate. We can also show how the limiting cases of flat space (constant curvature zero) and the two-dimensional hyperboloid (constant negative curvature) emerge.
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