Disorder and localization in ultracold quantum gases

In this work we study Anderson localization of quantum states in different kind of disordered media. First we apply the recently-developed theory of \textit{landscape localization} pioneered by Filoche and Mayboroda, giving rise to an effective potential which yields further insight into localization. We analyze this problem numerically in one, two and three dimensions with uniform and point-like potentials, aiming at inferring the properties of the localized states from the knowledge of the localization landscape. We also consider quasiperiodic potentials which induce a metal-insulator transition (extended to localized states), testing the prediction of the wave functions using the effective potential as a reference. Finally, we study how the localized states are distributed in the energy spectrum energy in one-dimension and how we can modify their localization varying the properties of the random potential. We find that for a uniform disordered potential, the density of states is symmetric around the mean energy. And for a point-like potential, the spectrum energy has a continuous part at low energies and a discrete part at high energies