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 $B \in L \mathrm {log} L (\Omega) \cap C^\alpha (\Omega_0)$ 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 $\lambda (t) = Ce^{-ct^\sigma}$ 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 $\mathbb{R}^2$.