Eigenvalues of the Magnetic Neumann Laplacian on Domains with Peaks

Abstract

We consider the magnetic Neumann Laplacian on bounded domains in $\mathbb{R}^2$ with outward peaks. The operator is associated with a large magnetic field depending on a parameter $λ$, and we investigate the behaviour of the eigenvalues as $λ$ tends to $+\infty$. We show that their asymptotic expansion is influenced by the sharpness $q$ and geometry of the peak, and that its main term is of order $λ^{\frac{2}{q+1}}$. This is an extension of previous works on magnetic Neumann Laplacians in smooth domains and domains with corners.