  We study a simple model of a two-dimensional s-wave superconductor in the
presence of a random potential in the regime of large disorder. We first use
the Bogoliubov-de Gennes (BdG) approach to show that, with increasing disorder
the pairing amplitude becomes spatially inhomogeneous, and the system cannot be
described within conventional approaches for studying disordered
superconductors which assume a uniform order parameter. In the high disorder
regime, we find that the system breaks up into superconducting islands (with
large pairing amplitude) separated by an insulating sea. We show that this
inhomogeneity has important implications for the physical properties of this
system, such as superfluid density and the density of states. We find that a
finite spectral gap persists in the density of states for all values of
disorder and we provide a detailed understanding of this remarkable result. We
next generalize Anderson's idea of the pairing of exact eigenstates to include
an inhomogeneous pairing amplitude, and show that it is able to qualitatively
capture many of the nontrivial features of the full BdG analysis. Finally, we
study the transition to a gapped insulating state driven by the quantum phase
fluctuations about the inhomogeneous superconducting state.
