# Crystal Ugarte: A numerical investigation of Anderson localization in weakly interacting Bose gases - A study on the Gross-Pitaevskii eigenvalue problem

Master thesis presentation in Scientific Computing

**Time: **
Fri 2020-01-31 09.00

**Location: **
KTH, F11

**Subject area: **
Scientific Computing

**Respondent: **
Crystal Ugarte

**Opponent: **
Lucas Pettersson

**Supervisor: **
Patrick Henning

### Abstract

The ground state of a quantum system is the minimizer of the total energy of that system. The aim of this thesis is to present and numerically solve the Gross-Pitaevskii eigenvalue problem (GPE) as a physical model for the formation of ground states of dilute Bose gases at ultra-low temperatures in a disorder potential. The first part of the report introduces the quantum mechanical phenomenon that arises at ground states of the Bose gases; the Anderson localization, and presents the nonlinear eigenvalue problem and the finite element method (FEM) used to discretize the GPE. The numerical method used to solve the eigenvalue problem for the smallest eigenvalue is called the inverse power iteration method, which is presented and explained. In the second part of the report, the smallest eigenvalue of a linear Schrödinger equation is compared with the numerically computed smallest eigenvalue (ground state) in order to evaluate the accuracy of a linear numerical scheme constructed as first step for numerically solving the non-linear problem. In the next part of the report, the numerical methods are implemented to solve for the eigenvalue and eigenfunction of the (non-linear) GPE at ground state (smallest eigenvalue). The mathematical expression for the quantum energy and smallest eigenvalue of the non-linear system are presented in the report and the FEM and inverse power iteration method are used to solve respective discrete equations.

As the aim is to evaluate the GPE as a physical model for the formation of Anderson localization, the following physical questions will be answered in this report; How does the Anderson localization depend on the disorder and strength of the potential field applied to the system? What happens to the Anderson localization phenomena when we change the number and the type of particles in it (change the chemical element)? For that reason, there is also an emphasis on quantum mechanical theory in the introduction and the results section of the report to answer the questions posed.