  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.
