Some new results about the massless Dirac operator

In this paper, we give a method to construct the zero mode and zero resonance of the massless Dirac operator H = α·D + Q(x) and obtain some special examples to show the sharpness of some existing results. On the other hand, we extend the previous results of H about the vanishing properties of the zero mode and the existence of the zero resonance. In Sec. IV, we consider the local version and get some improvements and extensions.