Abstract
This work is focused on the different supersymmetric extensions of the Landau model. We aim to fully solve each model and describe its energy levels, wavefunctions, Hilbert space and define a norm on it, as well as find symmetry generators and transformations with respect to them. Several possible generalizations were considered before. It was found for Landau model on the so called Superflag manifold as well as planar Superflag and Superplane Landau models that standard norm on the Hilbert space is not positive definite. Later for planar cases it was found that it is possible to fix this by introducing a new norm which will be invariant and positive definite. Surprisingly this procedure brings up "hidden" symmetries for the known super Landau models. In the dissertation we apply the same procedure for Landau model on superpshere and Superflag manifolds. It turns out that superpsherical Landau model is equivalent to the Superflag model with one of the parameters fixed. Because the model on superpshere can be recovered from the Superflag we will do calculations of corrected norm only for the Superflag. After this we develop a different generalization of the Superplane Landau model. Starting with Lagrangian in a superfield form we introduce two arbitrary functions of superfields K(Φ) and V(Φ) into the Lagrangian. We follow with the component form of Lagrangian. The quantization of the model is possible, and we will show that there is a reparametrization which turn equation of motion of the first scheme into the second set. Standard metric is again non-positive definite and we apply already known procedure to correct it. It will not be possible to solve Schrodinger equations in general with undefined K and V, so we consider one specific case which give us Landau model on a sphere with N = 2 supersymmetry, which put it apart from the superspherical Landau model, which have a superpshere for a target space but do not possess supersymmetry.