The Magnetic Laplacian with a Higher-order Vanishing Magnetic Field in a Bounded Domain

Abstract: This paper is concerned with spectrum properties of the magnetic Laplacian with a higher-order vanishing magnetic field in a bounded domain. We study the asymptotic behaviors of ground state energies for the Dirichlet Laplacian, the Neumann Laplacian, and the Dirichlet-to-Neumann operator, as the field strength parameter $\beta$ goes to infinite. Assume that the magnetic field does not vanish to infinite order, we establish the leading orders of $\beta$. We also obtain the first terms in the asymptotic expansions with remainder estimates under additional assumptions on an invariant subspace for a Taylor polynomial of the magnetic field. Our aim is to provide a unified approach to all three cases.