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arXiv:2502.20318 · Studies in Applied Mathematics

Non‐Relativistic Limit of Dirac Hamiltonians With Aharonov–Bohm Fields

Matteo Gallone, Alessandro Michelangeli, Diego Noja

Published Apr 1, 2026• 0 citations
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Abstract

ABSTRACT We characterize the families of self‐adjoint Dirac and Schrödinger operators with Aharonov–Bohm magnetic field, and we exploit the non‐relativistic limit of infinite light speed to connect the former to the latter. The limit consists of the customary removal of the rest energy and of a suitable scaling, with the light speed, of the short‐scale boundary condition of self‐adjointness. This ensures that the scattering length of the Aharonov–Bohm interaction is preserved along the limit. Noteworthy is the fact that the whole family of Dirac‐AB operators is mapped, in the non‐relativistic limit, into the physically relevant sub‐family of ‐wave, angular‐momentum‐commuting, Schrödinger–AB Hamiltonians with relativistic Dirac approximants.

DOI
10.1111/sapm.70209

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ABSTRACT We characterize the families of self‐adjoint Dirac and Schrödinger operators with Aharonov–Bohm magnetic field, and we exploit the non‐relativistic limit of infinite light speed to connect the former to the latter. The limit consists of the customary removal of the rest energy and of a suitable scaling, with the light speed, of the short‐scale boundary condition of self‐adjointness. This ensures that the scattering length of the Aharonov–Bohm interaction is preserved along the limit. Noteworthy is the fact that the whole family of Dirac‐AB operators is mapped, in the non‐relativistic limit, into the physically relevant sub‐family of ‐wave, angular‐momentum‐commuting, Schrödinger–AB Hamiltonians with relativistic Dirac approximants.
The formulation of Berry for the Aharonov-Bohm effect is generalized to the relativistic regime. Then, the problem of finding the self-adjoint extensions of the (2+1)-dimensional Dirac Hamiltonian, in an Aharonov-Bohm background potential, is solved in a novel way. The same treatment also solves the problem of finding the self-adjoint extensions of the Dirac Hamiltonian in a background Aharonov-Casher. PACS: 03.65.Bz, 03.65.Pm Typeset using REVTEX 1 A novel systematics for the computation of the energy eigenfunctions of a spinless electrically charged nonrelativistic particle in the presence of an Aharonov-Bohm [1] external potential, was put forward by Berry [2] sometime ago. We propose here a generalization of this systematics to the case where the particle is a relativistic electron. As will be shown, the set of eigenfunctions found through this generalization serves as a basis for writing any function in the domain of self-adjointness of the corresponding Dirac Hamiltonian. This corroborates the correctness of our proposal, which can also be used to obtain the selfadjoint extensions of the Dirac Hamiltonian in an Aharonov-Casher [3] background field. The extension of Berry’s ideas to the relativistic regime is the original contribution in this work. To make this paper self contained, we start by reviewing the main features of Berry’s formulation [2]. As known, the Hamiltonian operator H(⃗r, ⃗p) describing the dynamics of a spinless two-dimensional nonrelativistic particle (mass m and electric charge e) in the presence of an external static magnetic field B (B = ⃗ ∇ × ⃗ ( A) is H(⃗r, ⃗p) = H0 ⃗r, ⃗p − e ⃗ A(⃗r)) where ⃗r ≡ (x1, x2) are the particle coordinates, ⃗p ≡ (p1, p2) are the corresponding canonically conjugate momenta and H0(⃗r, ⃗p) is the free particle Hamiltonian. The energy eigenfunctions of the operator H can formally be constructed in terms of those of H0 as follows [4] ψ(⃗r) = ψ0(⃗r) exp ie ⃗r
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.
We study all the s.a. Schrodinger and Dirac operators (Hamiltonians) both with pure AB field and with magnetic-solenoid field. Then, we perform a complete spectral analysis for these operators, which includes finding spectra and spectral decompositions, or inversion formulas. In constructing the Hamiltonians and performing their spectral analysis, we respectively follow the von Neumann theory of s.a. extensions of symmetric differential operators and the Krein method of guiding functionals. The examples of similar consideration are given by us in arXiv:0903.5277, where a nonrelativistic particle in the Calogero potential field is considered and in Theor. Math. Phys. 150 (1) (2007) 34, where a Dirac particle in the Coulomb field of arbitrary charge is considered. However, due to peculiarities of the three-dimensional problems under consideration, we elaborated a generalization of the approach used in the study of the Dirac particle.
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.