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

TL;DR

This paper establishes lower bounds for the first eigenvalue of the Laplacian with zero magnetic field in multiply connected planar domains, linking spectral properties to geometric and topological features.

Contribution

It introduces new lower bounds for the eigenvalues based on domain geometry, topology, and magnetic fluxes, including sharp bounds for doubly connected domains.

Findings

Lower bounds depend on area, perimeter, diameter, fluxes, and width ratio.

Sharp lower bound for doubly connected domains.

Extension to eigenvalues of Aharonov-Bohm operators with multiple poles.

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.