  Numerical simulations suggest that merging double white dwarfs (WDs) may
produce a newborn neutron star surrounded by a fossil disk. We investigate the
evolution of the fossil disk following the coalescence of double WDs. We
demonstrate that the evolution can be mainly divided into four phases: the slim
disk phase (with time $\lesssim$ 1 yr), the inner slim plus outer thin disk
phase ($\sim 10-\DP{6}$ yr), the thin disk phase ($\sim \DP{2}-\DP{7}$ yr), and
the inner advection-dominated accretion flow plus outer thin disk phase, given
the initial disk mass $\sim 0.05-0.5\,M_{\sun}$ and the disk formation time
$10^{-3}-1$ s. Considering possible wind mass loss from the disk, we present
both analytic formulae and numerically calculated results for the disk
evolution, which is sensitive to the condition that determines the location of
the outer disk radius. The systems are shown to be very bright in X-rays in the
early phase, but quickly become transient within $\lesssim$ 100 yr, with peak
luminosities decreasing with time. We suggest that they might account for part
of the very faint X-ray transients around the Galactic center region, which
generally require a very low mass transfer rate.
