Full text
| |
| Author(s): |
Gustavo Barbagallo de Oliveira
Total Authors: 1
|
| Document type: | Master's Dissertation |
| Press: | São Paulo. |
| Institution: | Universidade de São Paulo (USP). Instituto de Física (IF/SBI) |
| Defense date: | 2004-07-15 |
| Examining board members: |
Domingos Humberto Urbano Marchetti;
Joao Carlos Alves Barata;
Cesar Rogerio de Oliveira
|
| Advisor: | Domingos Humberto Urbano Marchetti |
| Abstract | |
We considered a system of ordinary differential equations with a constant coefficient matrix A perturbed by a quasi-periodic matrix Q(t). We assumed that A and Q(t) are skew-Hermitian. In Physics, this is the Schrödinger equation for a two-level system in the presence of a potential quasi-periodic in time. In this dissertation we study the reducibility problem for this equation, that is, we investigate under which conditions on A and Q(t) the solution to this equation is quasi-periodic. In addition to the mathematical interest on the problem, this type of model have received attention recently in the study of \"quantum chaos\". Firstly, we will formulate the reducibilty problem in a general context and analyze two particular situations: periodic systems and scalar systems. Secondly, we will discuss some results about reducibility available in the literature. Finally, we will prove two theorems about reducibility for the above system using methods available in the literature. In the first theorem, we will use a KAM type procedure to control small denominators. In the second, we will combine the KAM procedure with the inverse function theorem. (AU) | |