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

2025/03/18 by Bernard, Helffer, Francois, Nicoleau · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2503.14008

Abstract

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

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• Low-lying eigenvalues in the semiclassical limit of a Schrödinger operator with an inverse square potential, and non-asymptotic a-zeros of Kummer functions
• Flux effects on Magnetic Laplace and Steklov eigenvalues in the exterior of a disk

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