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.
Highlights
We consider a smooth, connected and bounded domain Ω ⊂ R2 of area |Ω|
Let A be a closed 1-form and introduce the magnetic Neumann Laplacian ΔA with potential A acting on functions u ∈ C∞(Ω, C)
The following notation is sometimes used: ΔA = (i∇ + A♯)2, We take Neumann boundary conditions and we study the eigenvalue problem: ΔAu = u on Ω
Summary
We consider a smooth, connected and bounded domain Ω ⊂ R2 of area |Ω|. Let A be a closed 1-form and introduce the magnetic Neumann Laplacian ΔA with potential A acting on functions u ∈ C∞(Ω, C). For a more detailed introduction to the magnetic Neumann Laplacian associated to a closed potential, see the introduction of [5] and the references therein It is precisely the goal of this note to investigate how the topology and the geometry of the domain Ω influence the ground state energy 1(Ω, A) when the magnetic field is zero. The topology of a planar domain Ω is specified by the number n = n(Ω) of holes, and our first main result, Theorem 1, gives an upper bound of the ground state energy depending only on the area of Ω and the number of holes; up to a numerical constant, the bound is sharp and is achieved for a certain class of punctured domains (see Theorem 3).
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