  Near-horizon conformal structure of a massive Schwarzschild black hole of
mass M is analyzed using a scalar field as a simple probe of the background
geometry. The near-horizon dynamics is governed by an operator which is related
to the Virasoro algebra and admits a one-parameter family of self-adjoint
extensions described by a real parameter z. When z satisfies a suitable
contraint, the corresponding wavefunctions exhibit scaling behaviour in a
band-like region near the horizon of the black hole. This formalism is
consistent with the Bekenstein-Hawking entropy formula and naturally produces
the -3/2 log M^2 correction term to the black hole entropy with other
subleading corrections exponentially suppressed. This precise form for the
black hole entropy is expected on general grounds in any conformal field
theoretic description of the problem. The presence of the Virasoro algebra and
the scaling properties of the associated wavefunctions in the near-horizon
region together with the appearance of the logarithmic correction to the
Bekenstein-Hawking entropy provide strong evidence for the near-horizon
conformal structure in this system.
