Approximate Zero Modes for the Pauli Operator on a Region

Let ${\mathcal{P}}_{\mathrm{\Omega },tA}$ denoted the Pauli operator on a bounded open region $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with Dirichlet boundary conditions and magnetic potential $A$ scaled by some $t>0$. Assume that the corresponding magnetic field $B=\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}A$ satisfies $B\in L\log L(\Omega)\cap C^\alpha(\Omega_0)$where$\alpha>0$and$\Omega_0$isanopensubsetof$\Omega$offullmeasure(notethat,theOrliczspace$L\log L(\Omega)$contains$L^p(\Omega)$forany$p>1$)\mathrm{.}Let$$\mathsf{N}$_{\Omega,tA}(\lambda)$ denote the corresponding eigenvalue counting function. We establish the strong field asymptotic formula $${\mathsf{N}}_{\mathrm{\Omega },tA}(\lambda (t))=\frac{t}{2\pi }{\int }_{\mathrm{\Omega }}\mid B(x)\mid dx+o(t)$$ as $t\to +\mathrm{\infty }$, 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 ${\mathbb{R}}^2$.