# Approximate zero modes for the Pauli operator on a region

### Daniel M. Elton

Lancaster University, UK

## Abstract

Let $P_{Ω,tB}$ denoted the the Pauli operator on a bounded open region $Ω⊂R_{2}$ with Dirichlet boundary conditions and magnetic potential A scaled by some $t>0$. Assume that the corresponding magnetic field $B=curlA$ satisfies $B∈LlogL(Ω)∩C_{α}(Ω_{0})$ where $α>0$ and $Ω_{0}$ is an open subset of $Ω$ of full measure (note that, the Orlicz space $LgL(Ω)$ contains $L_{p}(Ω)$ for any $p>1$). Let $N_{Ω,tB}(λ)$ denote the corresponding eigenvalue counting function. We establish the strong field asymptotic formula

as $t→+∞$, whenever $λ(t)=Ce_{−ct_{σ}}$ for some $σ∈(0,1)$ 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 $R_{2}$.

## Cite this article

Daniel M. Elton, Approximate zero modes for the Pauli operator on a region. J. Spectr. Theory 6 (2016), no. 2, pp. 373–413

DOI 10.4171/JST/127