非常曲率空间上的量子力学：振子问题和超可积性
Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability
摘要：
严格求出了N维非常曲率空间上非线性振子的量子态的全谱和本征函数，由于其下的曲率空间产生了依赖于位置的动能，在量子化后必须保持系统的最大超可积性的条件下，给出了三种不同的量子化公式.通过适当的相似变换，详细描述了这三个薛定谔问题之间的关系。这三种方法被用来说明N维弯曲空间的量子化问题的不同特点，或者，位置相关的质量量子哈密顿。据我们所知，这个量子振子是在具有非常数曲率的N维空间上的最大超可积量子系统的第一个例子。
英文摘要：
The full spectrum and eigenfunctions of the quantum version of a nonlinear oscillator defined on an N-dimensional space with nonconstant curvature are rigorously found. Since the underlying curved space generates a position-dependent kinetic energy, three different quantization prescriptions are worked out by imposing that the maximal superintegrability of the system has to be preserved after quantization. The relationships among these three Schroedinger problems are described in detail through appropriate similarity transformations. These three approaches are used to illustrate different features of the quantization problem on N-dimensional curved spaces or, alternatively, of position-dependent mass quantum Hamiltonians. This quantum oscillator is, to the best of our knowledge, the first example of a maximally superintegrable quantum system on an N-dimensional space with nonconstant curvature.
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