Examples of resonant phenomena in varieties of macroscopic and microscopic systems are abundant in nature. These features are observable in a range of systems from ocean waves to Fermi and Bose gases in very sophisticated sate-of-the-art experiments. For past few decades, there have been a lot of developments in investigations of collective oscillations in semiconductor heterostructures, cold fermion and boson gases. In some cases theoretical predictions on the behavior of such oscillations predate the experimental evidences by decades, and in other cases there are still unexplained experimental results. In real experiments the system is always finite and confined by external forces which gives rise to many interesting behaviors and lately numerous theoretical works have also addressed the issue of confinement in such systems. This work addresses an important factor in theoretical considerations, namely the interparticle interactions.
In the first part of this thesis, I present a quantum-phenomenological calculation of the collective mode frequencies of a 1D interacting electron gas laterally confined by a parabolic potential under the influence of an external magnetic field, which will also be complemented by a perturbative solution. In the second part, I talk about the effect on finite non-parabolicity in the confinement potential. It will be showed that the collective oscillations of this nature are detuned by such irregularity and also that they have finite lifetime as opposed to the parabolic case. In the final part I discuss the collective modes in an interacting Bose-Einstein Condensate (BEC) coupled with its thermal cloud in a pancake-shaped confinement. The geometry is taken to be infinite in the radial direction and confined laterally with a parabolic potential. I will show how the coupled system behaves behaves in presence of interaction, and also how to incorporate non-parabolicty to the solution.