Non-monotonicity for the 3D magnetic Robin Laplacian

TL;DR

This paper presents one of the first counterexamples demonstrating non-monotonic behavior of the lowest eigenvalue for the 3D magnetic Robin Laplacian, highlighting differences from the 2D case.

Contribution

It provides a novel counterexample in three dimensions showing non-monotonicity of the lowest eigenvalue for the magnetic Robin Laplacian.

Findings

Counterexample in 3D case showing non-monotonicity

Non-monotonic eigenvalue asymptotics for Robin parameter tending to +∞

Differences from 2D magnetic Laplacian behavior

Abstract

Previous works provided several counterexamples to monotonicity of the lowest eigenvalue for the magnetic Laplacian in the two-dimensional case. However, the three-dimensional case is less studied. We use the results obtained by Helffer, Kachmar and Raymond to provide one of the first counterexamples in 3D. Considering the Robin magnetic Laplacian on the unit ball with a constant magnetic field, we show the non-monotonicity of the lowest eigenvalue asymptotics when the Robin parameter tends to $+\infty$.