On the magnetic Dirichlet to Neumann operator on the exterior of the disk – Diamagnetism, weak-magnetic field limit and flux effects

Abstract

In this paper, we analyze the magnetic Dirichlet-to-Neumann operator (D-to-N map) $\check{\Lambda }(b,\nu )$ on the exterior of the disk with respect to a magnetic potential $A_{b,\nu }=A^b+A_{\nu }$ where, for $b\in R$ and $\nu \in R$, $A^b(x,y)=b(-y,x)$ and $A_{\nu }(x,y)$ is the Aharonov-Bohm potential centered at the origin of flux $2\pi \nu$. First, we show that the limit of $\check{\Lambda }(b,\nu )$ as $b\to 0$ is equal to the D-to-N map $\hat{\Lambda }\left(\nu \right)$ on the interior of the disk associated with the potential $A_{\nu }(x,y)$. Secondly, we study the ground state energy of the D-to-N map $\check{\Lambda }(b,\nu )$ and show that the strong diamagnetism property holds. Finally we slightly extend to the exterior case the asymptotic results as $b\to \infty$ obtained in the interior case for general domains.