Aspects of hidden and manifest SL(2,R) symmetry in 2D near-horizon black-hole background

The invariance under unitary representations of the conformal group SL(2,R) of a quantum particle is rigorously investigated in two-dimensional spacetimes containing Killing horizons using DFF model. The limit of the near-horizon approximation is considered. If the Killing horizon is bifurcate the conformal symmetry is hidden, i.e. it does not arise from geometrical spacetime isometries, but the whole Hilbert space turns out to be an irreducible unitary representation of SL(2,R) and the time evolution is embodied in the unitary representation. In this case the symmetry does not depend on the mass of the particle and, if the representation is faithful, the conformal observable K shows thermal properties. If the Killing horizon is nonbifurcate the conformal symmetry is manifest, i.e. it arises from geometrical spacetime isometries. The SL(2,R) representation which arises from the geometry selects a hidden conformal representation. Also in that case the Hilbert space is an irreducible representation of SL(2,R) and the group conformal symmetries embodies the time evolution with respect to the local Killing time. However no thermal properties are involved. The conformal observable K gives rise to Killing time evolution of the quantum state with respect to another global Killing time present in the manifold. Mathematical proofs about the developed machinery are supplied and features of the operator H_g = -({d^2}/{dx^2})+ ({g}/{x^2}), with g=-1/4 are discussed. It is proven that a statement, used in the recent literature, about the spectrum of self-adjoint extensions of H_g is incorrect.