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Journal of Physics A Mathematical and General

Self-adjoint extensions of the Laplacian and Aharonov–Bohm operators with a potential supported on a circle

Mahouton Norbert Hounkonnou

Published Sep 23, 2003• 0 citations
Publisher

Abstract

Using the definition of a self-adjoint operator, we give the self-adjoint extensions of the Laplacian and Aharonov–Bohm operators with a potential supported on a circle of radius R . This study gives rise to interesting results, namely all self-adjoint extensions are given by a four-parameter family of self-adjoint operators, reproducing a matrix representation for U (2) symmetry.

DOI
10.1088/0305-4470/36/40/l01

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Using the definition of a self-adjoint operator, we give the self-adjoint extensions of the Laplacian and Aharonov–Bohm operators with a potential supported on a circle of radius R . This study gives rise to interesting results, namely all self-adjoint extensions are given by a four-parameter family of self-adjoint operators, reproducing a matrix representation for U (2) symmetry.
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 develop a general technique for finding self-adjoint extensions of a symmetric operator that respects a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schrödinger operators with singular potentials. The approach is based on constructing a unitary transformation diagonalizing the symmetries and reducing the initial operator to the direct integral of a suitable family of partial operators. We prove that symmetry preserving self-adjoint extensions of the initial operator are in a one-to-one correspondence with measurable families of self-adjoint extensions of partial operators obtained by reduction. The general scheme is applied to the three-dimensional Aharonov-Bohm Hamiltonian describing the electron in the magnetic field of an infinitely thin solenoid. We construct all self-adjoint extensions of this Hamiltonian, invariant under translations along the solenoid and rotations around it, and explicitly find their eigenfunction expansions.