Approximate zero modes for the Pauli operator on a region

  • Daniel M. Elton

    Lancaster University, UK


Let PΩ,tB\mathcal{P}_{\Omega,tB} denoted the the Pauli operator on a bounded open region ΩR2\Omega\subset\mathbb{R}^2 with Dirichlet boundary conditions and magnetic potential A scaled by some t>0t > 0. Assume that the corresponding magnetic field B=curlAB = \mathrm {curl} A satisfies BLlogL(Ω)Cα(Ω0)B \in L \mathrm {log} L (\Omega) \cap C^\alpha (\Omega_0) where α>0\alpha>0 and Ω0\Omega_0 is an open subset of Ω\Omega of full measure (note that, the Orlicz space LlogL(Ω)L\log L(\Omega) contains Lp(Ω)L^p(\Omega) for any p>1p>1). Let NΩ,tB(λ)\mathsf{N}_{\Omega,tB}(\lambda) denote the corresponding eigenvalue counting function. We establish the strong field asymptotic formula

NΩ,tA(λ(t))=t2πΩB(x)dx  +o(t)\mathsf{N}_{\Omega,tA}(\lambda(t))=\frac{t}{2\pi}\int_{\Omega} \lvert B(x) \rvert\, dx\;+o(t)

as t+t \to +\infty, whenever λ(t)=Cectσ\lambda (t) = Ce^{-ct^\sigma} for some σ(0,1)\sigma \in (0,1) and c,C>0c,C > 0. The corresponding eigenfunctions can be viewed as a localised version of the Aharonov–Casher zero modes for the Pauli operator on R2\mathbb{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