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arXiv:2608.00637

Remarks on Regular Approximations to the Robin Aharonov-Bohm Hamiltonian

Cesar R. de Oliveira

math-phmath.SPquant-phSubmitted Aug 1, 2026• Announced Aug 4, 2026
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Abstract

In the space $\mathbb R^3$, for Robin parameter $L>0$, it is shown that there is a family of Schrödinger operators with penetrable toroidal solenoid and variable conductivity that approximates the magnetic Aharonov-Bohm operator with a Robin boundary condition at the solenoid (border). It is also shown that approximations via smooth potentials and then a barrier in the solenoid interior give Dirichlet boundary conditions. The approximations are in the strong resolvent sense and obtained through the $Γ$-convergence technique, and they hold for the more general setting of smooth, closed and compact surfaces and continuous and bounded magnetic potentials.

DOI
10.1142/S0129055X26500157
Comments
Accepted in Reviews in Mathematical Physics

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In the space R3\mathbb R^3R3, for Robin parameter L>0L>0L>0, it is shown that there is a family of Schrödinger operators with penetrable toroidal solenoid and variable conductivity that approximates the magnetic Aharonov-Bohm operator with a Robin boundary condition at the solenoid (border). It is also shown that approximations via smooth potentials and then a barrier in the solenoid interior give Dirichlet boundary conditions. The approximations are in the strong resolvent sense and obtained through the ΓΓΓ-convergence technique, and they hold for the more general setting of smooth, closed and compact surfaces and continuous and bounded magnetic potentials.
We consider the Hamiltonian operator associated with planar sections of infinitely long cylindrical solenoids and with a homogeneous magnetic field in their interior.First, in the Sobolev space H 2 , we characterize all generalized boundary conditions on the solenoid border compatible with quantum mechanics, i.e. the boundary conditions, so that the corresponding Hamiltonian operators are self-adjoint.Then we study and compare the scattering of the most usual boundary conditions, that is, Dirichlet, Neumann and Robin.
We add a scalar potential to the 2D Aharonov-Bohm (AB) model which properly diverges both at the solenoid border and at infinity so that the resulting operator is essentially self-adjoint and has a discrete spectrum; the former property is interpreted as no contact of the particle with the solenoid border since there is no need of boundary conditions. We study gauge transformations to get the usual periodic behavior of the AB properties as a function of the magnetic flux. The presence of the AB effect is proven through the ground state energy, which is shown to be smooth in case it is simple and with a nonzero derivative if the ground state is real valued; such properties are verified in the case of circular solenoids, for which it is shown to be a nonconstant periodic function with a minimum at integer and a maximum at half-integer circulations (at half-integer circulations, it is doubly degenerated).
It is well known that the formal Aharonov–Bohm Hamiltonian operator, describing the interaction of a charged particle with a magnetic vortex, has a four-parameter family of self-adjoint extensions, which reduces to a two-parameter family if one requires that the Hamiltonian commutes with the angular momentum operator. The question we study here is which of these self-adjoint extensions can considered as limits of regularized Aharonov–Bohm Hamiltonians, that is Pauli Hamiltonians in which the magnetic field corresponds to a flux tube of nonzero diameter. We show that not all the self-adjoint extensions in this two-parameter family can be obtained by these approximations, but only two one-parameter subfamilies. In these two cases we can choose the gyromagnetic ratio in the approximating Pauli Hamiltonian in such a way that we get convergence in the norm resolvent sense to the corresponding self-adjoint extension.
The charged particle scattering amplitude on the magnetic field of a toroidal solenoid is obtained in the first Born and high-energy approximations. It is shown that multiconnectedness of the accessible space, non-triviality vector potentials in it and the single-valuedness of the wavefunctions used are not enough for the existence of the Aharonov-Bohm effect. Criteria are given for this. Surrounding the toroidal solenoid with barriers of different geometrical forms, the author observes that the magnetic field contribution to the scattering amplitude depends crucially both on the barrier height and its form. The latter could hardly be explained by particle penetration into the H not=0 region. The author proposes an addendum to Tonomura's experiments which probably could clear up any doubts about the existence of the AB effect. The gedanken experiment is discussed which shows that the effect of inaccessible fields could be observed even in a simply connected space.