Single-particle density of states, bound states, phase-shift flip, and a resonance in the presence of an Aharonov-Bohm potential

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Abstract
Both the nonrelativistic scattering and the spectrum in the presence of the Aharonov-Bohm potential are analyzed and the single-particle density of states for different self-adjoint extensions is calculated. The single-particle density of states is shown to be a symmetric and periodic function of the flux, which depends only on the distance from the nearest integer. The Krein-Friedel formula for this long-range potential is shown to be valid when regularized with the ζ function. The limit when the radius R of the flux tube shrinks to zero is discussed. For R≠0 and in the case of an anomalous magnetic moment gm≳2 (note, e.g., that gm=2.002 32 for the electron) the coupling for spin-down electrons is enhanced and bound states occur in the spectrum. Their number does depend on a regularization and generically does not match with the number of zero modes in a given field that occur when gm=2. Provided the coupling with the interior of the flux tube is not renormalized to a critical one, neither bound states nor zero modes survive the limit R→0. The Aharonov-Casher theorem on the number of zero modes is corrected for the singular field configuration.