  Accretion disks can be eccentric: they support $m=1$ modes that are global
and slowly precessing. But whether the modes remain trapped in the disk---and
hence are long-lived---depends on conditions at the outer edge of the disk.
Here we show that in disks with realistic boundaries, in which the surface
density drops rapidly beyond a given radius, eccentric modes are trapped and
hence long-lived. We focus on pressure-only disks around a central mass, and
show how this result can be understood with the help of a simple second-order
WKB theory. We show that the longest lived mode is the zero-node mode in which
all of the disk's elliptical streamlines are aligned, and that this mode decays
coherently on the viscous timescale of the disk. Hence such a mode, once
excited, will live for the lifetime of the disk. It may be responsible for
asymmetries seen in recent images of protoplanetary disks.
