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Approximate Zero Modes for the Pauli Operator on a Region

Let $\mathcal{P}_{Ω,tA}$ denoted the Pauli operator on a bounded open region $Ω\subset\mathbb{R}^2$ with Dirichlet boundary conditions and magnetic potential $A$ scaled by some $t>0$. Assume that the corresponding magnetic field $B=\mathrm{curl}\,A$ satisfies $B\in L\log L(Ω)\cap C^α(Ω_0)$ where $α>0$ and $Ω_0$ is an open subset of $Ω$ of full measure (note that, the Orlicz space $L\log L(Ω)$ contains $L^p(Ω)$ for any $p>1$). Let $\mathsf{N}_{Ω,tA}(λ)$ denote the corresponding eigenvalue counting function. We establish the strong field asymptotic formula \[ \mathsf{N}_{Ω,tA}(λ(t))=\frac{t}{2π}\int_Ω\lvert B(x)\rvert\,dx\;+o(t) \] as $t\to+\infty$, whenever $λ(t)=Ce^{-ct^σ}$ for some $σ\in(0,1)$ 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 $\mathbb{R}^2$.