  We study quantum systems on a discrete bounded lattice (lattice billiards).
The statistical properties of their spectra show universal features related to
the regular or chaotic character of their classical continuum counterparts.
However, the decay dynamics of the open systems appear very different from the
continuum case, their properties being dominated by the states in the band
center. We identify a class of states ("lattice scars") that survive for
infinite times in dissipative systems and that are degenerate at the center of
the band. We provide analytical arguments for their existence in any bipartite
lattice, and give a formula to determine their number. These states should be
relevant to quantum transport in discrete systems, and we discuss how to
observe them using photonic waveguides, cold atoms in optical lattices, and
quantum circuits.
