  We develop an inhomogeneous quantum mean-field theory for disordered
particle-hole symmetric Bose-Hubbard models in two dimensions. Collective
excitations are described by fluctuations about the mean-field ground state. In
quadratic (Gaussian) approximation, the Goldstone (phase) and Higgs (amplitude)
modes completely decouple. Each is described by a disordered Bogoliubov
Hamiltonian which can be solved by an inhomogeneous multi-mode Bogoliubov
transformation. We find that the Higgs modes are noncritical and strictly
localized everywhere in the phase diagram. In contrast, the lowest-energy
Goldstone mode delocalizes in the superfluid phase. We discuss these findings
from the perspective of conventional Anderson localization theory. We also
compare the effects of different types of disorder such as site dilution and
random interactions; we relate our results to recent quantum Monte Carlo
simulations, and we discuss the limits and generality of our approach.
