Aharonov–Bohm quantum systems on a punctured 2-torus

Abstract

This paper deals with Aharonov–Bohm (A-B) quantum systems on a punctured two-dimensional torus from the geometric and the operator theoretic point of view. First, flat connections on the U(1)-bundles over the punctured 2-torus are studied, which serve as vector potentials for A-B effect magnetic fields. It is proved that the moduli space of flat connections is identified with the (N + 1)-dimensional torus TN+1, if the punctured torus has N > 0 pinholes at which solenoids are assumed to penetrate the 2-torus. For a given point of TN+1, an associated flat connection is constructed in terms of the Weierstrass zeta function on C together with an inhomogeneous linear function on R2. A-B quantum systems are defined in terms of position operators and momentum operators coupled with the A-B potentials. Necessary and sufficient conditions are given for two A-B quantum systems to be unitarily equivalent. Further, the A-B Hamiltonian is defined and analysed from the viewpoint of operator theory. The deficiency indices of the A-B Hamiltonian are determined to be (N + M, N + M), where M is the number of solenoids whose fluxes are not quantized. Finally, the eigenvalue problem is studied for the A-B Hamiltonian with all fluxes quantized to obtain eigenvalues together with eigenfunctions which are described in terms of the Weierstrass sigma functions.