Flux effects on Magnetic Laplace and Steklov eigenvalues in the exterior of a disk

TL;DR

This paper derives a detailed asymptotic expansion for the lowest eigenvalue of magnetic Laplace and Steklov operators outside a disk, highlighting flux dependence in strong and weak magnetic field limits.

Contribution

It provides a three-term asymptotic expansion for eigenvalues, improving prior results and explicitly revealing flux dependence in the asymptotics.

Findings

Three-term asymptotic expansion for eigenvalues in strong magnetic fields

Explicit flux dependence encoded in the third term of the expansion

Analysis of flux effects in weak magnetic field limit

Abstract

We derive a three-term asymptotic expansion for the lowest eigenvalue of the magnetic Laplace and Steklov operators in the exterior of the unit disk in the strong magnetic field limit. This improves recent results of Helffer-Nicoleau (2025) based on special function asymptotics, and extends earlier works by Fournais-Helffer (2006), Kachmar (2006), and R. Fahs, L. Treust, N. Raymond, S. V\~u Ng\d{o}c (2024). Notably, our analysis reveals how the third term encodes the dependence on the magnetic flux. Finally, we investigate the weak magnetic field limit and establish the flux dependence in the asymptotics of Kachmar-Lotoreichik-Sundqvist (2025).