Asymptotics for the magnetic Dirichlet-to-Neumann eigenvalues in general domains

TL;DR

This paper investigates the behavior of magnetic Dirichlet-to-Neumann eigenvalues in general domains, proving conjectures about their divergence with increasing magnetic field and extending results to three dimensions and related operators.

Contribution

It provides the first rigorous proof of the conjecture for general 2D domains and extends the analysis to 3D, linking eigenvalue asymptotics with magnetic Robin Laplacian.

Findings

Eigenvalues tend to infinity as magnetic field increases in 2D domains.

Refined asymptotic estimates are established for general domains.

Connections with magnetic Robin Laplacian eigenvalues are explored.

Abstract

Inspired by a paper by T. Chakradhar, K. Gittins, G. Habib and N. Peyerimhoff, we analyze their conjecture that the ground state energy of the magnetic Dirichlet-to-Neumann operator tends to infinity as the magnetic field tends to infinity. More precisely, we prove refined conjectures for general two dimensional domains, based on the analysis in the case of the half-plane and the disk by two of us (B.H. and F.N.). We also extend our analysis to the three dimensional case, and explore a connection with the eigenvalue asymptotics of the magnetic Robin Laplacian.