Asymptotics for eigenvalues of one-dimensional Dirac operators in the weak coupling limit

TL;DR

This paper investigates the asymptotic behavior of eigenvalues for one-dimensional Dirac operators under weak coupling, providing new asymptotic formulas for different classes of potentials using variational and resolvent methods.

Contribution

It introduces novel asymptotic results for eigenvalues of Dirac operators in the weak coupling limit, covering both Hermitian and non-Hermitian potentials with new analytical techniques.

Findings

Derived leading asymptotic term for Hermitian potentials with decay |V(x)| ~ |x|^{-1}

Obtained second asymptotic term for non-Hermitian L^1 potentials under moment conditions

Applied min-max and Birman-Schwinger principles to analyze eigenvalue behavior

Abstract

In this paper, we derive new results on the asymptotic behavior of eigenvalues of perturbed one-dimensional massive Dirac operators in the weak coupling limit. Two classes of potentials are considered. For bounded Hermitian potentials $V$ satisfying $|V(x)| \lesssim |x|^{-1}$ for large $|x|$, we recover the leading term, which may include a logarithmic correction if $V(x) \sim |x|^{-1}$ at infinity. For possibly non-Hermitian $L^1$ potentials satisfying a suitable moment condition, we obtain the second term in the asymptotic expansion. The first result is based on a min-max principle adapted to the non-relativistic limit, while the second result is obtained via the Birman-Schwinger principle and resolvent expansions.