Friedel sum rule and Levinson theorem for massless Weyl fermions

The Friedel sum rule for massless Weyl fermions is given. Due to the parity violation of Weyl fermions it is shown that the global phases of spinors are important in determining the accurate sum rule, and the Levinson theorem is proved to be determined only by the number of discrete zero modes. For potential barriers that do not allow any zero modes it is revealed that a particular duality relation exists between the phase shifts at the Fermi surface and the critical point of zero momentum.