Virtual bound states of the Pauli operator with an Aharonov-Bohm potential
Marie Fialova, David Krejcirik
TL;DR
This work analyzes the two-dimensional Pauli operator under a singular Aharonov–Bohm field, showing that both spin components are critical for α ∈ (0,1) and that arbitrarily small attractive electric perturbations produce two virtual bound states with polynomial-in-ε leading behavior. The authors deploy extension theory to rigorously define H_α, compute the Green function via Krein’s formula, and apply the Birman–Schwinger principle to reduce the weak-coupling problem to a finite-dimensional matrix equation, enabling explicit asymptotics. A key finding is that the AB singularity yields two distinct zero-energy resonances and that the weakly coupled eigenvalues scale as z_+(ε) ∼ −(-ε a_ε)^{1/(1−α)} and z_−(ε) ∼ −(-ε b_ε)^{1/α} in the diagonal perturbation case, with constants a_ε,b_ε determined by moments of V. The results also indicate that the AB-driven criticality cannot in general be approximated by regular potentials, highlighting delicate operator-limit behavior, while the framework accommodates complex perturbations and matrix-valued V. Overall, the paper provides a precise spectral-theoretic treatment of virtual bound states in a highly singular magnetic setting, with clear implications for zero-energy resonance phenomena and flux-tapped spin interactions in two dimensions.
Abstract
A maximal realisation of the two-dimensional Pauli operator, subject to Aharonov--Bohm magnetic field, is investigated. Contrary to the case of the Pauli operator with regular magnetic potentials, it is shown that both components of the Pauli operator are critical. Asymptotics of the weakly coupled eigenvalues, generated by electric (not necessarily self-adjoint) perturbations, are derived.