The EMS Publishing House is now EMS Press and has its new home at

Please find all EMS Press journals and articles on the new platform.

Journal of Spectral Theory

Full-Text PDF (372 KB) | Metadata | Table of Contents | JST summary
Volume 6, Issue 2, 2016, pp. 373–413
DOI: 10.4171/JST/127

Published online: 2016-04-06

Approximate zero modes for the Pauli operator on a region

Daniel M. Elton[1]

(1) Lancaster University, UK

Let $\mathcal{P}_{\Omega,tB}$ denoted the the Pauli operator on a bounded open region $\Omega\subset\mathbb{R}^2$ with Dirichlet boundary conditions and magnetic potential$ $A scaled by some $t > 0$. Assume that the corresponding magnetic field $B = \mathrm {curl} A$ satisfies $B \in L \mathrm {log} L (\Omega) \cap C^\alpha (\Omega_0)$ where $\alpha>0$ and $\Omega_0$ is an open subset of $\Omega$ of full measure (note that, the Orlicz space $L\log L(\Omega)$ contains $L^p(\Omega)$ for any $p>1$). Let $\mathsf{N}_{\Omega,tB}(\lambda)$ denote the corresponding eigenvalue counting function. We establish the strong field asymptotic formula $$\mathsf{N}_{\Omega,tA}(\lambda(t))=\frac{t}{2\pi}\int_{\Omega} \lvert B(x) \rvert\, dx\;+o(t)$$ as $t \to +\infty$, whenever $\lambda (t) = Ce^{-ct^\sigma}$ for some $\sigma \in (0,1)$ and $c,C > 0$. The corresponding eigenfunctions can be viewed as a localised version of the Aharonov–Casher zero modes for the Pauli operator on $\mathbb{R}^2$.

Keywords: Pauli operator, eigenvalue asymptotics, approximate zero modes

Elton Daniel: Approximate zero modes for the Pauli operator on a region. J. Spectr. Theory 6 (2016), 373-413. doi: 10.4171/JST/127