  We study the time-dependent occupation of an impurity state hybridized with a
continuum of extended or localized states. Of particular interest is the return
probability, which gives the long-time limit of the average impurity
occupation. In the extended case, the return probability is zero unless there
are bound states of the impurity and continuum. We present exact expressions
for the return probability of an impurity state coupled to a lattice, and show
that the existence of bound states depends on the dimension of the lattice. In
a disordered lattice with localized eigenstates, the finite extent of the
eigenstates results in a non-zero return probability. We investigate different
parameter regimes numerically by exact diagonalization, and show that the
return probability can serve as a measure of the localization length in the
regime of weak hybridization and disorder. Possible experimental realizations
with ultracold atoms are discussed.
