  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.
