Approximate zero modes for the Pauli operator on a region

Abstract

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 $B\in L\log L(\Omega )\cap C^{\alpha }({\Omega }_0)$ 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

as $t\to +\infty$, whenever $\lambda (t)=C{e}^{-c{t}^{\sigma }}$ 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 ${\mathbb{R}}^2$.