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In this paper, we show that massless Dirac waves in the Schwarzschildgeometry decay to zero at a rate $t^{-2\lambda}$, where $\lambda=1, 2,...$ isthe angular momentum. Our technique is to use Chandrasekhar's separation ofvariables 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 thesecond 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 byshowing that the coefficient of the explicit profile is exactly zero. The keyingredients in the proof of the decay rate of solutions for the first set ofwave equations are an energy estimate used to show the absence of bound statesand zero energy resonance and the analysis of the spectral representation ofthe solutions. The proof of asymptotic behavior for the solutions of the secondset of wave equations relies on careful analysis of the Green's functions fortime independent Schr\"odinger equations associated with these wave equations.

Asymptotic Behavior of Massless Dirac Waves in Schwarzschild geometry