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Journal of Physics A Mathematical and General

Self-adjoint extensions of the Hamiltonian for a charged spin- 1/2 particle in the Aharonov-Bohm field

Francisco Antônio Bezerra Coutinho, Yukihisa Nogami, J. Fernando Perez, F. Masafumi Toyama

Published Oct 7, 1994• 12 citations
Publisher

Abstract

Recently we studied self-adjoint extensions (SAEs) of the Aharonov-Bohm Hamiltonian for a charged Schrodinger particle of spin 0. In this paper we discuss the relation between the SAEs for a spinless particle and those for a spin- 1/2 particle in the same environment. We also consider SAEs when a Coulomb potential is added. We furthermore clarify a few confusing issues which have been discussed in some recent papers regarding the relativistic Dirac particle in the Aharonov-Bohm field.

DOI
10.1088/0305-4470/27/19/024

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Recently we studied self-adjoint extensions (SAEs) of the Aharonov-Bohm Hamiltonian for a charged Schrodinger particle of spin 0. In this paper we discuss the relation between the SAEs for a spinless particle and those for a spin- 1/2 particle in the same environment. We also consider SAEs when a Coulomb potential is added. We furthermore clarify a few confusing issues which have been discussed in some recent papers regarding the relativistic Dirac particle in the Aharonov-Bohm field.
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
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.
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.