On localization of eigenfunctions of the magnetic Laplacian

2023/08/30 by Ovall, Jeffrey S., Quan, Hadrian, Reid, Robyn +1
#Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2308.15994

Abstract

Let Ω⊂ ℝ^d and consider the magnetic Laplace operator given by H(A) = (- i∇ - A(x))², where A:Ω→ ℝ^d, subject to Dirichlet eigenfunction. This operator can, for certain vector fields A, have eigenfunctions H(A) ψ= λψ that are highly localized in a small region of Ω. The main goal of this paper is to show that if |ψ| assumes its maximum in x₀ ∈ Ω, then A behaves `almost' like a conservative vector field in a 1/√λ-neighborhood of x₀ in a precise sense: we expect localization in regions where |curl A | is small. The result is illustrated with numerical examples.

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