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

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.

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