Title: Quantum mechanics on a punctured two-torus

Abstract:
The Aharonov-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 Aharonov and Bohm, and
verified by Tonomura {\it 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 master thesis, 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 are 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 {\bf C} together with
an inhomogeneous linear function on ${\bf R}^2$.
Since the flat connections serve as vector potentials for
A-B effect magnetic fields,
quantum systems describing the A-B effect 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.