Lower bounds for the first eigenvalue of the Laplacian with zero magnetic field in planar domains

Abstract

We study the Laplacian with zero magnetic field acting on complex functions of a planar domain $\Omega$, with magnetic Neumann boundary conditions. If $\Omega$ is simply connected then the spectrum reduces to the spectrum of the usual Neumann Laplacian; therefore we focus on multiply connected domains bounded by convex curves and prove lower bounds for its ground state depending on the geometry and the topology of $\Omega$. Besides the area, the perimeter and the diameter, the geometric invariants which play a crucial role in the estimates are the the fluxes of the potential one-form around the inner holes and the distance between the boundary components of the domain; more precisely, the ratio between its minimal and maximal width. Then, we give a lower bound for doubly connected domains which is sharp in terms of this ratio, and a general lower bound for domains with an arbitrary number of holes. When the inner holes shrink to points, we obtain as a corollary a lower bound for the first eigenvalue of the so-called Aharonov-Bohm operators with an arbitrary number of poles.