Studies Of Breathing Mode On A One-dimensional Cold Atomic System Based On Density Functional Theory

In this paper, the breathing modes on two-component fermi gases under harmonic confinements are studied by using the Gaudin-Yang model of homoge-neous systems. We perform a numerical study by using density functional theory (DFT)based on the Bethe-ansatz solution and a local density approximation (L-DA). The density profile and breathing modes are studied based on the above mentioned methods.In chapter1, we describe the background of cold atoms and the low di-mensional system, as well as the knowledge of the one-dimensional strongly correlated systems. At the same time, the methods for numerical solution of one-dimensional strongly correlated systems are also introduced, for instance, the Bethc-ansatz method. Moreover we introduce the one-dimensional tight-binding Fermi gas models:the one-dimensional Hubbard model and the one-dimensional Gaudin-Yang model.In chapter2, we introduce the methods of studying the breathing modes on Low-dimensional cold atomic systems:the local density approximation and density functional theory based on the Bcthe-ansatz solution. Moreover, the advantages and disadvantages of are discussed.In chapter3, firstly we use the density functional theory to obtain the density profile of two-component fermi gases under harmonic confinements, compared a-gainst the ones from the local density approximation. For a repulsive interaction, density functional theory gives more accurate ground-state density distributions of Fricdcl oscillation, which makes the breathing mode slightly smaller than the local density approximation results. However, for the weak attractive interac-tion, the density functional theory gives more oscillating results, which makes the breathing modes very different from the local density approximation ones. For the strong attractive interaction, the density functional theory based on the Bethe-ansatz is incorrect. In this case, the breathing modes deviate from the lo-cal density approximation ones. Finally we illustrate, when the interaction tends to zero, and the system is the ideal Fermi gas. In the case, the frequency of the breathing mode is4, when the interaction approaches infinity, equivalent to the completely Fermi gas. the frequency of the breathing mode also is4.In chapter4, we give conclusions and outlook.