  The axial modes for non-barotropic relativistic rotating neutron stars with
uniform angular velocity are studied, using the slow-rotation formalism
together with the low-frequency approximation, first investigated by Kojima.
The time independent form of the equations leads to a singular eigenvalue
problem, which admits a continuous spectrum. We show that for $l=2$, it is
nevertheless also possible to find discrete mode solutions (the $r$-modes).
However, under certain conditions related to the equation of state and the
compactness of the stellar model, the eigenfrequency lies inside the continuous
band and the associated velocity perturbation is divergent; hence these
solutions have to be discarded as being unphysical. We corroborate our results
by explicitly integrating the time dependent equations. For stellar models
admitting a physical $r$-mode solution, it can indeed be excited by arbitrary
initial data. For models admitting only an unphysical mode solution, the
evolutions do not show any tendency to oscillate with the respective frequency.
For higher values of $l$, it seems that in certain cases there are no mode
solutions at all.

General Relativity and Quantum Cosmology;Astrophysics