穿过Grushin奇点的量子粒子
Quantum particle across Grushin singularity
摘要：
一类模型被认为是约束在退化的黎曼流形称为Grushin圆柱体，并自由移动的量子粒子只受潜在的几何形状：相应的谱分析的详细开发，鉴于传输的奇异性，分离的两个半圆柱体的现象。而经典的对应物总是由一个粒子在有限时间内沿着测地线落在度规的奇点轨迹上组成，量子模型可能会显示几何限制，或者相反的部分透射和反射。所有的本地实现的自由（拉普拉斯-贝尔特拉米）量子哈密顿量检查的非等效协议的传输/反射和其光谱的结构的特点，包括适用时，他们的基态和积极性。此外，还进行了稳态散射分析，计算了透射系数和反射系数。这使得理解所谓的“桥接”传输协议先前确定的文献中，我们恢复和我们的系统分析内的研究的杰出地位。
英文摘要：
A class of models is considered for a quantum particle constrained on degenerate Riemannian manifolds known as Grushin cylinders, and moving freely subject only to the underlying geometry: the corresponding spectral analysis is developed in detail in view of the phenomenon of transmission across the singularity that separates the two half-cylinders. Whereas the classical counterpart always consists of a particle falling in finite time along the geodesics onto the metric's singularity locus, the quantum models may display geometric confinement, or on the opposite partial transmission and reflection. All the local realisations of the free (Laplace-Beltrami) quantum Hamiltonian are examined as non-equivalent protocols of transmission/reflection and the structure of their spectrum is characterised, including when applicable their ground state and positivity. Besides, the stationary scattering analysis is developed and transmission and reflection coefficients are calculated. This allows to comprehend the distinguished status of the so-called `bridging' transmission protocol previously identified in the literature, which we recover and study within our systematic analysis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
