This paper studies the nature of fractional linear transformations in a general relativity context as well as in a quantum theoretical framework. Two features are found to deserve special attention: the first is the possibility of separating the limit-point condition at infinity into loxodromic, hyperbolic, parabolic and elliptic cases. This is useful in a context in which one wants to look for a correspondence between essentially self-adjoint spherically symmetric Hamiltonians of quantum physics and the theory of Bondi–Metzner–Sachs transformations in general relativity. The analogy therefore arising suggests that further investigations might be performed for a theory in which the role of fractional linear maps is viewed as a bridge between the quantum theory and general relativity. The second aspect to point out is the possibility of interpreting the limit-point condition at both ends of the positive real line, for a second-order singular differential operator, which occurs frequently in applied quantum mechanics, as the limiting procedure arising from a very particular Kleinian group which is the hyperbolic cyclic group. In this framework, this work finds that a consistent system of equations can be derived and studied. Hence, one is led to consider the entire transcendental functions, from which it is possible to construct a fundamental system of solutions of a second-order differential equation with singular behavior at both ends of the positive real line, which in turn satisfy the limit-point conditions. Further developments in this direction might also be obtained by constructing a fundamental system of solutions and then deriving the differential equation whose solutions are the independent system first obtained. This guarantees two important properties at the same time: the essential self-adjointness of a second-order differential operator and the existence of a conserved quantity which is an automorphic function for the cyclic group chosen.

中文翻译：

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广义相对论和量子力学中的分数线性映射

本文研究了广义相对论和量子理论框架中分数线性变换的性质。发现有两个特征值得特别注意：第一个是将无穷远处的极限点条件分解为等距、双曲线、抛物线和椭圆情况的可能性。这在人们想要寻找量子物理学的基本自伴球对称哈密顿量与广义相对论中的邦迪-梅茨纳-萨克斯变换理论之间的对应关系时很有用。因此出现的类比表明，可以对分数线性映射的作用被视为量子理论和广义相对论之间的桥梁的理论进行进一步的研究。需要指出的第二个方面是，对于在应用量子力学中经常出现的二阶奇异微分算子，将正实线两端的极限点条件解释为由非常特殊的 Kleinian 群是双曲循环群。在这个框架下，这项工作发现可以推导出和研究一致的方程组。因此，人们被引导考虑整个超越函数，从中可以构造一个在正实线两端具有奇异行为的二阶微分方程的基本解系统，这反过来又满足极限-点条件。在这个方向上的进一步发展也可以通过构造一个基本的解系统，然后推导出其解是首先获得的独立系统的微分方程来获得。这同时保证了两个重要的性质：二阶微分算子的本质自伴随性和作为所选循环群的自守函数的守恒量的存在。