The Monodromy Matrix for a Family of Almost Periodic Schr Odinger Equations in the Adiabatic Case

This work is devoted to the study of a family of almost periodic one-dimensional Schrr odinger equations. We deene a monodromy matrix for this family. We study the asymptotic behavior of this matrix in the adiabatic case. Therefore, we develop a complex WKB method for adiabatic perturbations of periodic Schrr odinger equations. At last, the study of the monodromy matrix enables us to get some spectral results for the initial family of almost periodic equations. R esum e. Ce travail est consacr e a l' etude d'une famille d' equations de Schrr odin-ger quasi-p eriodiques en dimension 1. Nous d eenissons une matrice de mo-nodromie pour cette famille. Nous etudions le comportement asymptotique de cette matrice dans le cas adiabatique. A cette n, nous d eveloppons une version de la m ethode WKB complexe pour l' equation de Schrr odinger p eriodique avec une perturbation adiabatique. Ennn, l' etude de la matrice de monodromie nous permet d'obtenir des r esultats spectraux pour la famille d' equations quasi-p eriodiques initiale. 1. Introduction and statement of the main results In this paper, we study the family of Schrr odinger equations d 2 dx 2 (x) + (V (x) + W ("x)) (x) = E (x); x 2 R: (1.1) Here 2 R is a parameter numbering the equations, V and W are two periodic functions satisfying " is a xed positive number. The ratio of the periods of the potential in (1.1) is equal to 2=". If 2=" 6 2 Q , the potential V (x) + W ("x) is almost periodic. Our ultimate goal is to understand the spectral properties of the family (1.1) assuming that V is a real-valued measurable bounded function of x 2 R and that W is analytic in 2 C. 1 It is well known that for one-dimensional periodic Schrr odinger equation, the key to the understanding of its spectral properties is the monodromy matrix. It is not possible to deene this object for a single almost periodic diierential equation. In this paper, we will show that, for a family of almost periodic diierential equations of type (1.1), a monodromy matrix can be deened and used for its spectral analysis. Of course, a thorough understanding of the structure of monodromy matrix is crucial. Therefore, in this paper, we will need to take the adiabatic limit " ! 0. As we shall see, the monodromy matrix always …