  In this paper we construct a solvable toy model of the quantum dynamics of
the interior of a spherical black hole with falling spherical scalar field
excitations. We first argue about how some aspects of the quantum gravity
dynamics of realistic black holes emitting Hawking radiation can be modelled
using Kantowski-Sachs solutions with a massless scalar field when one focuses
on the deep interior region $r\ll M$ (including the singularity). Further, we
show that in the $r\ll M$ regime, and in suitable variables, the KS model
becomes exactly solvable at both the classical and quantum levels. The quantum
dynamics inspired by loop quantum gravity is revisited. We propose a natural
polymer-quantization where the area $a$ of the orbits of the rotation group is
quantized. The polymer (or loop) dynamics is closely related with the
Schroedinger dynamics away from the singularity with a form of continuum limit
naturally emerging from the polymer treatment. The Dirac observable associated
to the mass is quantized and shown to have an infinite degeneracy associated to
the so-called $\epsilon$-sectors. Suitable continuum superpositions of these
are well defined distributions in the fundamental Hilbert space and satisfy the
continuum Schroedinger dynamics.
