Title: Quantum mechanics on a punctured two-torus

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.