The asymptotic limits of zero modes of massless Dirac operators

TL;DR

This paper investigates the asymptotic behavior of zero modes of the massless Dirac operator with decaying potentials, showing that their scaled limits exist and are characterized by an integral involving the potential and the zero mode.

Contribution

It establishes the existence of asymptotic limits for zero modes of the massless Dirac operator with decaying Hermitian matrix potentials, linking the limits to integral expressions.

Findings

The limit of |x|^2 times the zero mode exists as |x| approaches infinity.

The asymptotic limit is expressed via an integral involving the potential and the zero mode.

Zero modes exhibit specific asymptotic behavior determined by the decay rate of the potential.

Abstract

Asymptotic behaviors of zero modes of the massless Dirac operator $H=\alpha\cdot D + Q(x)$ are discussed, where $\alpha= (\alpha_1, \alpha_2, \alpha_3)$ is the triple of $4 \times 4$ Dirac matrices, $ D=\frac{1}{i} \nabla_x$, and $Q(x)=\big(q_{jk} (x) \big)$ is a $4\times 4$ Hermitian matrix-valued function with $| q_{jk}(x) | \le C < x >^{-\rho} $, $\rho >1$. We shall show that for every zero mode $f$, the asymptotic limit of $|x|^2f(x)$ as $|x| \to +\infty$ exists. The limit is expressed in terms of an integral of $Q(x)f(x)$.