Approximate Zero Modes for the Pauli Operator on a Region

Let $\mathcal{P}_{\Omega,tA}$ denoted 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 $\mathsf{N}_{\Omega,tA}(\lambda)$ 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 $\mathbb{R}^2$.