Upper bounds for the ground state energy of the Laplacian with zero magnetic field on planar domains

TL;DR

This paper derives upper bounds for the first eigenvalue of the magnetic Laplacian with zero magnetic field on planar domains, relating it to domain topology and geometry, and refines these bounds using domain modifications.

Contribution

It provides new upper bounds for the magnetic Laplacian's ground state energy based on domain topology, with sharpness results and refinements via domain cuts.

Findings

Upper bounds depend on the ratio of holes to area.

Bounds are sharp and attained by Aharonov-Bohm-type operators.

Refined bounds involve Cheeger-type constants and domain modifications.

Abstract

We obtain upper bounds for the first eigenvalue of the magnetic Laplacian associated to a closed potential $1$-form (hence, with zero magnetic field) acting on complex functions of a planar domain $\Omega$, with magnetic Neumann boundary conditions. It is well-known that the first eigenvalue is positive whenever the potential admits at least one non-integral flux. By gauge invariance the lowest eigenvalue is simply zero if the domain is simply connected; then, we obtain an upper bound of the ground state energy depending only on the ratio between the number of holes and the area; modulo a numerical constant the upper bound is sharp and we show that in fact equality is attained (modulo a constant) for Aharonov-Bohm-type operators acting on domains punctured at a maximal $\epsilon$-net. In the last part we show that the upper bound can be refined, provided that one can transform the given domain in a simply connected one by performing a number of cuts with sufficiently small total length; we thus obtain an upper bound of the lowest eigenvalue by the ratio between the number of holes and the area, multiplied by a Cheeger-type constant, which tends to zero when the domain is metrically close to a simply connected one.