We consider the strong field asymptotics for the occurrence of zero modes of certain Weyl-Dirac operators on $mathbb{R}^3$. In particular we are interested in those operators $mathcal{D}_{B}$ for which the associated magnetic field $B$ is given by pu
lling back a $2$-form $beta$ from the sphere $mathbb{S}^2$ to $mathbb{R}^3$ using a combination of the Hopf fibration and inverse stereographic projection. If $int_{mathbb{S}^2}beta eq0$ we show that [ sum_{0le tle T}mathrm{dim},mathrm{Ker},mathcal{D}_{tB} =frac{T^2}{8pi^2},biggllvertint_{mathbb{S}^2}betabiggrrvert,int_{mathbb{S}^2}lvert{beta}rvert+o(T^2) ] as $Tto+infty$. The result relies on ErdH{o}s and Solovejs characterisation of the spectrum of $mathcal{D}_{tB}$ in terms of a family of Dirac operators on $mathbb{S}^2$, together with information about the strong field localisation of the Aharonov-Casher zero modes of the latter.
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 $mathbb{R}^2$.
We study the spectrum of a one-dimensional Dirac operator pencil, with a coupling constant in front of the potential considered as the spectral parameter. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupli
ng constant for which the kernel of the Dirac operator contains a square integrable function. In physics literature such a function is called a confined zero mode. Several results on the asymptotic distribution of coupling constants giving rise to zero modes are obtained. In particular, we show that this distribution depends in a subtle way on the sign variation and the presence of gaps in the potential. Surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential. We further observe that variable sign potentials may produce complex eigenvalues of the operator pencil. Some examples and numerical calculations illustrating these phenomena are presented.
