American Institute of Mathematical Sciences

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May  2008, 9(3&4, May): 541-554. doi: 10.3934/dcdsb.2008.9.541

On the spectrum of the one-dimensional Schrödinger operator

Nguyen Dinh Cong 1, and  Roberta Fabbri 2,

1. 

Institute of Mathematics, Vietnamese Academy of Science and Technology, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam
2. 

Università di Firenze, Dipartimento di Sistemi e Informatica, Via Santa Marta 3, 50139 Firenze

Received  January 2007 Revised  July 2007 Published  February 2008

The spectral theory of the one-dimensional Schrödinger operator with a quasi-periodic potential can be fruitfully studied considering the corresponding differential system. In fact the presence of an exponential dichotomy for the system is equivalent to the statement that the energy $E$ belongs to the resolvent of the operator. Starting from results already obtained for the spectrum in the continuous case, we show that in the discrete case a generic bounded measurable Schrödinger cocycle has Cantor spectrum.
Keywords: Quasi-periodic Schrödinger operator, Cantor spectrum, exponential dichotomy..
Mathematics Subject Classification: Primary: 34D08, 34D09; Secondary: 37B5.
Citation: Nguyen Dinh Cong, Roberta Fabbri. On the spectrum of the one-dimensional Schrödinger operator. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 541-554. doi: 10.3934/dcdsb.2008.9.541
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