Denysenko O. On the stability and instability zones of one-dimensional quasiperiodic Shrodinger equation

This thesis is devoted to study stability and instability zones for one-dimensional Shrodinger equation with smooth quasiperiodic potential. With a help of KAM-theory methods Shrodinger equation whose potential is characterized by certain rate of approximation by trigonometric polynomials or derivatives rise rate was investigated. There was described the set of energy, for which this equation has a pair of linearly independent Floquet-Bloch solutions. KAM-methods are applied to construct boundaries of instability zones and solutions for these boundaries. Instability zones' boundaries have been proved to be analytic in a small parameter. Estimates of resonance energy zones were obtained. The cases where the potential is quasianalytic, analytic and a finite order trigonometric polynomial were also considered. Diagram technique was applied to construct the boundaries of instability zones as convergent series in a small parameter.