  We use linear perturbation theory to investigate how a groove in the phase
space of a disc galaxy changes the stellar disc's stability properties. Such a
groove is a narrow trough around a fixed angular momentum from which most stars
have been removed, rendering part of the disc unresponsive to spiral waves. We
find that a groove can dramatically alter a disc's eigenmode spectrum by giving
rise to a set of vigorously growing eigenmodes. These eigenmodes are particular
to the grooved disc and are absent from the original ungrooved disc's mode
spectrum. We discuss the properties and possible origin of the different
families of new modes.
  By the very nature of our technique, we prove that a narrow phase-space
groove can be a source of rapidly growing spiral patterns that are true
eigenmodes of the grooved disc and that no non-linear processes need to be
invoked to explain their presence in N-body simulations of disc galaxies. Our
results lend support to the idea that spiral structure can be a recurrent
phenomenon, in which one generation of spiral modes alters a disc galaxy's
phase space in such a way that a following generation of modes is destabilized.
