Deficiency indices of Aharonov-Bohm Hamiltonian on a two-torus

YABU Yoshiro

27th Jan. (Thu) 13:00-

The Aharanov-Bohm effect (abbrev. A-B effect) is known as a topological effect that gives rise to an observable phase shift of a wave function. This effect was predicted by Aharanov and Bohm, and verified by Tonomura et al. by using the elaborate and precise experiment. After the discovery of the A-B effect, many topological concepts, like instanton, monopole and anyon, have been found to make mathematical physics fruitful.
In this study, the A-B effect on a punctured two-dimensional torus is intensively worked out from the geometric and the operator theoretic points of view. First, flat connections on the U(1)-bundles over the punctured two-dimensional torus and dealt with. It is proved that the moduli space of flat connections is identified with the (N+1)-dimensional torus T^{N+1}, if the punctured torus has N > 0 pinholes. For a given point of T^{N+1}, an associated flat connection is made up in terms of the Weierstrass zeta function on \mathbf{C} together with an inhomogeneous linear function on \mathbf{R}^2. Since the flat connections serve as vector potentials for A-B effect magnetic fields, quantum systems describing the A-B efffect on the punctured torus are then studied and classified. Necessary and sufficient conditions are given for two A-B effect systems to be unitarily equivalent. Finally, the A-B Hamiltonian is analyzed from the viewpoint of operator theory. It is shown that the essential self-adjointness of the Hamiltonian is equivalent to flux quantization of solenoid sitting at each pinhole. Further, eigenvalues of the essentially self-adjoint Hamiltonian are explicitly obtained together with eigenfunctions described in terms of the Weierstrass sigma functions.

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