Stability of Nonradial Vibrational Modes of Relativistic Neutron Stars

We establish an analytical connection between the amplitudes of gravitational waves scattered by a relativistic neutron star and its nonradial vibrational modes. The procedure is similar to the one followed by Jost and other authors who recognized that the bound-state Schrödinger wave functions in potential theory are analytically connected with the $S$ function for the scattering. In the case of potential theory, the $S$ function for the scattering is analytic in the wave number $k$, except for singular points or branch lines. The resonant and antiresonant states are represented by poles of $S$ in the lower half Gauss $k$ plane. The bound states, on the other hand, correspond to poles on the positive imaginary axis. Similarly, in the case of a neutron star the amplitudes of gravitational waves scattered by the star are shown, in the present paper, to be analytic in the wave circular frequency $\omega$. The stable and unstable vibrational modes of the neutron star give rise to resonances in the scattering of gravitational waves described by poles of scattering amplitude lying, respectively, in the upper and lower half Gauss $\omega$ plane. In particular, we obtain a result (similar to Levinson's theorem for potential scattering) which relates the unstable modes to scattering phase shifts.