Abstract 
In the present thesis I have studied problems which are related to the behaviour of
bosonic atoms at zero temperature and as a result they are BoseEinstein condensed.
Quite generally my thesis explores some of the effects which belong to the collection
of phenomena that constitute “superfluidity”.
In all my projects I have assumed onedimensional motion of the atoms, and I have
also imposed periodic boundary conditions, which is suitable for a ring potential. A
substantial part of my thesis focuses on the corrections due to the finiteness of the
atom number. These corrections come from correlations which show up beyond the
meanfield approximation. The derived results are based mostly on the method of
diagonalization of the manybody Hamiltonian, while I have also used the meanfield
approximation.
The novelty of my results relies on the combined effect of onedimensional motion,
the imposed periodic boundary conditions, and the small atom numbers that I considered.
As shown below, all of these give rise to effects which have not been investigated
so far.
The experimental motivation for my studies comes from numerous experiments
which have created and observed persistent currents in atomic BoseEinstein condensates
in topologicallynontrivial traps, i.e., annular and toroidal. In addition, the
advances in atom detection has allowed experimentalists to lower the number of
atoms and even work with just a few of them.
In the first project of my thesis I investigated two questions. The first was the phenomenon
of hysteresis, i.e., the hysteresis loop and the corresponding critical frequencies.
The second question was the critical coupling for stability of persistent
currents, paying particular attention to the effect of the finiteness of the atom number
on it.
In the second project of my thesis I studied the effect of the finiteness of the atom
number on the solitarywave solutions, going beyond the meanfield approximation.
To attack this problem, I developed a general strategy, and considered a linear
superposition of the eigenstates of the manybody Hamiltonian, with amplitudes
that I extracted from the mean field approximation. The resulting manybody state
has all the desired features and is lower in energy than the corresponding meanfield
state. In the third project I studied the rotational properties of a twocomponent Bose
Einstein condensed gas of distinguishable atoms. I demonstrated that the angular
momentum may be given to the system either via singleparticle, or “collective" excitation.
Finally, despite the complexity of this problem, under rather typical conditions
the excitation spectrum has a remarkably simple and regular form.
