  We investigate the dynamics of scalar fields in the near-horizon exterior
region of a Schwarzschild black hole. We show that low-energy modes are
typically long-living and might be considered as being confined near the black
hole horizon. Such dynamics are effectively governed by a Schroedinger operator
with infinitely many self-adjoint extensions parameterized by $U(1)$, a
situation closely resembling the case of an ordinary free particle moving on a
semiaxis. Even though these different self-adjoint extensions lead to
equivalent scattering and thermal processes, a comparison with a simplified
model suggests a physical prescription to chose the pertinent self-adjoint
extensions. However, since all extensions are in principle physically
equivalent, they might be considered in equal footing for statistical analyses
of near-horizon modes around black holes. Analogous results hold for any
non-extremal, spherically symmetric, asymptotically flat black hole.
