In this thesis we investigate the trapping of spin-polarized electrons in edge states around a pair of antidots (Paper I). This study supports a proposal for using the trapped electrons to realize quantum gates - the building blocks of a quantum computer. The main advantage of our proposal is that the edge states have a very long lifetime, which will reduce problems with decoherence.

We also address the issue of dimensionality by studying the local density of states in a quantum point contact (QPC)_. This is important since many results, regarding electron transport through the QPC_, rely on an assumption of one­ dimensionality. We show that in order for this assumption to be valid, certain conditions regarding the shape of the potential have to be fulfilled (Paper II).

Furthermore, we study electron transport in quantum wires, with emphasis on electron-electron interaction effects. In Papers III and IV we provide an explanation, based on these effects, of the experimentally observed 0.7 analogues, using Density Functional Theory (DFT).