Reduction by symmetries in singular quantum-mechanical problems: general scheme and application to Aharonov-Bohm model

TL;DR

This paper introduces a general method for constructing symmetry-preserving self-adjoint extensions of symmetric operators, applied specifically to the Aharonov-Bohm Hamiltonian, enabling explicit eigenfunction expansions.

Contribution

The paper develops a unified scheme for finding symmetry-respecting self-adjoint extensions and applies it to the Aharonov-Bohm model, explicitly characterizing all such extensions.

Findings

Constructed all symmetry-invariant self-adjoint extensions of the Aharonov-Bohm Hamiltonian.

Established a one-to-one correspondence between extensions of the original operator and reduced partial operators.

Derived explicit eigenfunction expansions for the extended Hamiltonians.

Abstract

We develop a general technique for finding self-adjoint extensions of a symmetric operator that respect a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schr\"odinger 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.