Helical magnetic fields and semi-classical asymptotics of the lowest eigenvalue

TL;DR

This paper derives precise asymptotic formulas for the lowest eigenvalue of a 3D Neumann magnetic Laplacian with a semi-classical parameter and non-uniform magnetic field, revealing geometric and magnetic influences.

Contribution

It provides a sharp two-term asymptotic expansion for the lowest eigenvalue, including effects of magnetic field geometry and domain shape, especially for helical fields in a unit ball.

Findings

Two-term asymptotics for the lowest eigenvalue established

Eigenfunction concentration occurs at symmetric points on the sphere

Results highlight the influence of magnetic field geometry on spectral properties

Abstract

We study the 3D Neuman magnetic Laplacian in the presence of a semi-classical parameter and a non-uniform magnetic field with constant intensity. We determine a sharp two term asymptotics for the lowest eigenvalue, where the second term involves a quantity related to the magnetic field and the geometry of the domain. In the special case of the unit ball and a helical magnetic field, the concentration takes place on two symmetric points of the unit sphere.