  We study the Anderson localization of Bogoliubov quasiparticles (elementary
many-body excitations) in a weakly interacting Bose gas of chemical potential
$\mu$ subjected to a disordered potential $V$. We introduce a general mapping
(valid for weak inhomogeneous potentials in any dimension) of the Bogoliubov-de
Gennes equations onto a single-particle Schr\"odinger-like equation with an
effective potential. For disordered potentials, the Schr\"odinger-like equation
accounts for the scattering and localization properties of the Bogoliubov
quasiparticles. We derive analytically the localization lengths for correlated
disordered potentials in the one-dimensional geometry. Our approach relies on a
perturbative expansion in $V/\mu$, which we develop up to third order, and we
discuss the impact of the various perturbation orders. Our predictions are
shown to be in very good agreement with direct numerical calculations. We
identify different localization regimes: For low energy, the effective
disordered potential exhibits a strong screening by the quasicondensate density
background, and localization is suppressed. For high-energy excitations, the
effective disordered potential reduces to the bare disordered potential, and
the localization properties of quasiparticles are the same as for free
particles. The maximum of localization is found at intermediate energy when the
quasicondensate healing length is of the order of the disorder correlation
length. Possible extensions of our work to higher dimensions are also
discussed.
