Path integral approach for superintegrable potentials on spaces of nonconstant curvature: I. Darboux spaces D _I and D _II
Path integral approach for superintegrable potentials on spaces of nonconstant curvature: I. Darboux spaces D
_I and D
_II
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非常曲率空间上超可积势的路径积分方法：I. 达布空间 D _I 和 D _II
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
A. Sissakian
中科院分区：
文献类型：
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作者：
C. Grosche;G. Pogosyan;A. Sissakian
本文将Feynman路径积分技巧应用于二维非常曲率空间上的超可积势：这些空间是Darboux空间D_I和D_II，D_I上有三个，D_II上有四个这样的势。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表述，因为它要么由涉及抛物线柱面函数的超越方程(达布空间I)确定，要么由高阶多项式方程确定。特别是D_I上的解表明，超可积系统不一定退化。我们还可以展示平面空间(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。
In this paper, the Feynman path integral technique is applied for superintegrable potentials on two-dimensional spaces of nonconstant curvature: these spaces are Darboux spaces D _I and D _II. On D _I, there are three, and on D _II four such potentials. 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.