We investigate the properties of r-modes characterized by the 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, does not have a discrete mode as a solution of the regular eigenvalue problem for polytropes with polytropic index N > 1.18, even in the post-Newtonian order. Furthermore, for an N = 1 polytrope, which is employed as a typical neutron star model, discrete r-mode solutions for the regular eigenvalue problem do not exist for stars whose relativistic factor M/R is larger than about 0.1, where M and R are stellar mass and stellar radius, respectively.
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