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On localization of eigenfunctions of the magnetic Laplacian
- Publication Year :
- 2023
-
Abstract
- Let $\Omega \subset \mathbb{R}^d$ and consider the magnetic Laplace operator given by $ H(A) = \left(- i\nabla - A(x)\right)^2$, where $A:\Omega \rightarrow \mathbb{R}^d$, subject to Dirichlet eigenfunction. This operator can, for certain vector fields $A$, have eigenfunctions $H(A) \psi = \lambda \psi$ that are highly localized in a small region of $\Omega$. The main goal of this paper is to show that if $|\psi|$ assumes its maximum in $x_0 \in \Omega$, then $A$ behaves `almost' like a conservative vector field in a $1/\sqrt{\lambda}-$neighborhood of $x_0$ in a precise sense: we expect localization in regions where $\left|\mbox{curl} A \right|$ is small. The result is illustrated with numerical examples.
- Subjects :
- Mathematics - Analysis of PDEs
Mathematics - Spectral Theory
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2308.15994
- Document Type :
- Working Paper