Explicit construction of quasi-periodic discrete schr\'odinger operators with cantor spectrum

We construct 1-dim difference Schr\"odinger operators with a class of Geverey potentials such that Cantor spectrum occurs together with the estimations of open spectral gaps. The proof is based on KAM and Moser-P\"oschel argument .
Comment: The loopholes in the proof of Chapter 5 and the errors in the text of Chapter 1 have been revised