离散半直线上的量子力学边界条件
Boundary conditions in quantum mechanics on the discretized half-line
摘要：
我们研究离散半线上的非相对论性量子力学，构造了一个单参数的哈密顿量族，它类似于连续半线量子力学中的Robin族边界条件.对于经典奇异汉密尔顿，该构造提供了一种奇异性避免机制，该机制与圈量子引力中遇到的奇异性避免具有定性相似之处。应用包括自由粒子，吸引库仑势，标度不变势和黑洞描述的Einstein-Rosen虫洞喉。频谱分析的解析和数值技术。在连续极限中，通过适当的微调可以恢复全部Robin族边界条件，但是Dirichlet型边界条件出现为通用的。
英文摘要：
We investigate nonrelativistic quantum mechanics on the discretized half-line, constructing a one-parameter family of Hamiltonians that are analogous to the Robin family of boundary conditions in continuum half-line quantum mechanics. For classically singular Hamiltonians, the construction provides a singularity avoidance mechanism that has qualitative similarities with singularity avoidance encountered in loop quantum gravity. Applications include the free particle, the attractive Coulomb potential, the scale invariant potential and a black hole described in terms of the Einstein-Rosen wormhole throat. The spectrum is analyzed by analytic and numerical techniques. In the continuum limit, the full Robin family of boundary conditions can be recovered via a suitable fine-tuning but the Dirichlet-type boundary condition emerges as generic.
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