Universal critical $g$ factor for spin-1/2 Aharonov-Bohm bound states

Abstract:Whether the spin-1/2 Aharonov-Bohm Hamiltonian supports bound states has remained controversial since Hagen concluded that no such states exist for a Dirac particle. Here, we show that Hagen's solution corresponds to a particular member of the one-parameter family of self-adjoint extensions that describe the singular Zeeman interaction, and we identify which member the microscopic physics selects. Matching the extension parameter to a finite-radius flux tube yields $\nu=0$ precisely at $g=2$, and it does so in every flux sector. The critical $g$ factor is therefore universal, $g_c=2$, and depends on neither the flux sector nor the regularization radius. The flux sector controls the depth of the bound state energy, and we obtain a closed-form expression showing that binding deepens markedly with the integer part of the flux. Hagen's conclusion is thus confirmed and sharpened: it is not a statement about one arbitrary member of a family of extensions, but about the member that the physics selects when the magnetic moment takes its Dirac value, and an anomalous moment is a necessary and sufficient condition for binding in the singular channel.

arXiv:2610.08521 [quant-ph]

(or arXiv:2610.08521v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.08521 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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