The Asymptotic Limits of Zero Modes of Massless Dirac Operators

Abstract

Asymptotic behaviors of zero modes of the massless Dirac operator H = α · D + Q(x) are discussed, where α = (α1, α2, α3) is the triple of 4 × 4 Dirac matrices, $$D = \frac{1}{i } \nabla_x$$ , and Q(x) = (q jk (x)) is a 4 × 4 Hermitian matrix-valued function with | q jk (x) | ≤ C 〈x〉−ρ, ρ > 1. We shall show that for every zero mode f, the asymptotic limit of |x|2 f (x) as |x| → + ∞ exists. The limit is expressed in terms of the Dirac matrices and an integral of Q(x) f (x).