Symmetry preserving self-adjoint extensions of Schr\"odinger operators with singular potentials

TL;DR

This paper introduces a method to find self-adjoint extensions of Schr"odinger operators with singular potentials that preserve symmetries, with applications to the Aharonov-Bohm Hamiltonian, enhancing understanding of quantum systems with singularities.

Contribution

It develops a general technique for symmetry-preserving self-adjoint extensions of Schr"odinger operators with singular potentials, applicable to complex quantum systems.

Findings

Established a one-to-one correspondence between symmetry-preserving extensions and families of partial operator extensions.

Applied the method to the Aharonov-Bohm Hamiltonian, demonstrating its practical utility.

Provided a systematic approach for analyzing quantum operators with singularities and symmetries.

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 construction is applied to the three-dimensional Aharonov-Bohm Hamiltonian describing the electron in the magnetic field of an infinitely thin solenoid.