Asymptotic Behavior of Massless Dirac Waves in Schwarzschild geometry

In this paper, we show that massless Dirac waves in the Schwarzschild geometry decay to zero at a rate $t^{-2\lambda}$, where $\lambda=1, 2,...$ is the angular momentum. Our technique is to use Chandrasekhar's separation of variables whereby the Dirac equations split into two sets of wave equations. For the first set, we show that the wave decays as $t^{-2\lambda}$. For the second set, in general, the solutions tend to some explicit profile at the rate $t^{-2\lambda}$. The decay rate of solutions of Dirac equations is achieved by showing that the coefficient of the explicit profile is exactly zero. The key ingredients in the proof of the decay rate of solutions for the first set of wave equations are an energy estimate used to show the absence of bound states and zero energy resonance and the analysis of the spectral representation of the solutions. The proof of asymptotic behavior for the solutions of the second set of wave equations relies on careful analysis of the Green's functions for time independent Schr\"odinger equations associated with these wave equations.