Nonlinear Analysis: Real World Applications 3 (2002) 37 – 59 www.elsevier.com/locate/na On the nature of the spectrum of the quasi-periodic Schr odinger operator Roberta Fabbri a ; * , Russell Johnson a , Raaella Pavani b a Dipartimento di Sistemi e Informatica, Universit a di Firenze, Via S. Marta 3, 50139 Firenze, Italy b Dipartimento di Matematica, Politecnico di Milano, P.za L. da Vinci 32, 20133 Milano, Italy Received 27 January 2000; accepted 4 November 2000 Keywords: Quasi-periodic Schrodinger operator; Cantor spectrum; Rotation number 1. Introduction The purpose of this paper is to study certain aspects of the one-dimensional Schr odinger operator with a quasi-periodic potential. We will present both theoret- ical and numerical results. As will be discussed in more detail below, our results complement those of the other papers on the subject in the sense that they are not of perturbative type, and make no a priori restrictions on the range of energies for which they are valid. The quasi-periodic Schr odinger operator has been intensively studied in the last 25 years, in both its discrete and its continuous versions. The discrete version is of basic importance in the theory of electron conduction in incommensurate media [33,2,1]. It is the point of departure for the study of the Hofstadter buttery [17], and for the development of the theory of the integral quantum Hall eect [22]. The continuous Schr odinger operator has a well-known connection to the Korteweg–de Vries equa- tion of water-wave theory, and most of what is known about the K–dV equation with quasi-periodic initial data can be traced to facts concerning the quasi-periodic Schr odinger operator [7,31]. Thus there is ample motivation for studying both the dis- crete and continuous quasi-periodic Schr odinger operators. * Corresponding author. E-mail addresses: fabbri@math.uni.it (R. Fabbri), johnson@ing1.ing.uni.it (R. Johnson). 1468-1218/01/$ - see front matter c 2001 Elsevier Science Ltd. All rights reserved. PII:S1468-1218(01)00012-8