We investigate properties of r -modes characterized by regular eigenvalue problem in slowly rotating relativistic polytropes .
Our numerical results suggest that discrete r -mode solutions for the regular eigenvalue problem exist only for restricted polytropic models .
In particular the r -mode associated with l = m = 2 , which is considered to be the most important for gravitational radiation driven instability , do not have a discrete mode as solutions of the regular eigenvalue problem for polytropes having the polytropic index N > 1.18 even in the post-Newtonian order .
Furthermore for a N = 1 polytrope , which is employed as a typical neutron star model , discrete r -mode solutions for regular eigenvalue problem do not exist for stars whose relativistic factor M / R is larger than about 0.1 .
Here M and R are stellar mass and stellar radius , respectively .