Journal of Spectral Theory

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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