On Localization of Eigenfunctions of The Magnetic Laplacian

• Published in: Reports on mathematical physics Vol. 94; no. 2; pp. 235 - 257
• Main Authors: Ovall, Jeffrey S., Quan, Hadrian, Reid, Robyn, Steinerberger, Stefan
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.10.2024
• Subjects: eigenfunction; localization; regularization; Schrödinger operator; localization; eigenfunction; Schrödinger operator; regularization
• ISSN: 0034-4877
• DOI: 10.1016/S0034-4877(24)00078-8

Let Ω ⊂ ℝd and consider the magnetic Laplace operator given by H(A) = (–i∇ – A(x))2, where A : Ω → ℝd, subject to Dirichlet boundary conditions. For certain vector fields A, this operator can 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 at x0 ∈ Ω, then A behaves 'almost' like a conservative vector field in a 1/λ-neighborhood of x0 in a precise sense. In particular, we expect localization in regions where |curl A| is small. The result is illustrated with numerical examples.