Approximate Zero Modes for the Pauli Operator on a Region

Let $mathcal{P}_{Omega,tA}$ denoted the Pauli operator on a bounded open region $Omegasubsetmathbb{R}^2$ with Dirichlet boundary conditions and magnetic potential $A$ scaled by some $t>0$. Assume that the corresponding magnetic field $B=mathrm{curl},A$ satisfies $Bin Llog L(Omega)cap C^alpha(Omega_0)$ where $alpha>0$ and $Omega_0$ is an open subset of $Omega$ of full measure (note that, the Orlicz space $Llog L(Omega)$ 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}{2pi}int_{Omega}lvert B(x)rvert,dx;+o(t) ] as $tto+infty$, whenever $lambda(t)=Ce^{-ct^sigma}$ for some $sigmain(0,1)$ 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 $mathbb{R}^2$.