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

Journal Article · Tue Dec 15 04:00:00 EST 2015 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4936305· OSTI ID:22479561

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.

• OSTI ID: 22479561
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 12 Vol. 56; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
AHARONOV-BOHM EFFECT
EIGENFUNCTIONS
HAMILTONIANS
MAGNETIC FIELDS
QUANTUM MECHANICS
ROTATION
SOLENOIDS
SYMMETRY