  Scale-free disks have no preferred length or time scale. The question has
been raised whether such disks have a continuum of unstable linear modes or
perhaps no unstable modes at all. We resolve this paradox by analysing the
particular case of a gaseous, isentropic disk with a completely flat rotation
curve (the Mestel disk) exactly. The heart of the matter is this: what are the
correct boundary conditions to impose at the origin or central cusp? We argue
that the linear stability problem is ill-posed. From any finite radius, waves
reach the origin after finite time but with logarithmically divergent phase.
Instabilities exist, but their pattern speeds depend upon an undetermined phase
with which waves are reflected from the origin. For any definite choice of this
phase, there is an infinite but discrete set of growing modes. Similar
ambiguities may afflict general disk models with power-law central cusps. The
ratio of growth rate to pattern speed, however, is independent of the central
phase. This ratio is derived in closed form for non self-gravitating normal
modes. The ratio for self-gravitating normal modes is found numerically by
solving recurrence relations in Mellin-transform space.
