| 摘要 | 第4-5页 |
| Abstract | 第5页 |
| Chapter 1 Introduction | 第9-13页 |
| 1.1 Discrete Schrodinger operators | 第9页 |
| 1.2 Motivation and Background | 第9-13页 |
| Chapter 2 Quantitative continuity of singular continuous spectral measuresand arithmetic criteria | 第13-59页 |
| 2.1 Introduction | 第13-27页 |
| 2.1.1 Main application | 第17页 |
| 2.1.2 Spectral singularity, continuity and proof of Theorem 2.1.4 | 第17-20页 |
| 2.1.3 Relation with other dimensions;Corollaries for the AMO,S-turmian potentials, and Transport exponents | 第20-24页 |
| 2.1.4 Preliminaries | 第24-27页 |
| 2.2 Spectral Continuity | 第27-44页 |
| 2.2.1 Proof of Theorem 2.1.6 | 第27-31页 |
| 2.2.2 Proof of Theorem 2.2.1 | 第31-32页 |
| 2.2.3 The hyperbolic case: Proof of Lemma 2.7 | 第32-37页 |
| 2.2.4 Energies with Trace close to 2: Proof of Lemma 2.8 | 第37-43页 |
| 2.2.5 Proof of Lemmas 2.5 and 2.6 | 第43-44页 |
| 2.3 Spectral Singularity | 第44-51页 |
| 2.3.1 Power law estimates and proof of Theorem 2.1.5 | 第44-48页 |
| 2.3.2 Proof of the density lemmas | 第48-51页 |
| 2.4 Sturmian Hamiltonian | 第51-53页 |
| 2.5 Appendix 1: Proof of Lemma 2.10: | 第53-54页 |
| 2.6 Appendix2: Proof of Lemma 2.13 and Lemma 2.14 | 第54-55页 |
| 2.7 Appendix3: Some estimates on matrix products | 第55-56页 |
| 2.8 Appendix4: Extended Schnol's Theorem (Lemma 2.9) | 第56-59页 |
| Chapter 3 Holder Continuity of the Lyapunov Exponent for Analytic Quasiperi-odic Schrodinger Cocycle with Weak Liouville Frequency | 第59-81页 |
| 3.1 Introduction and the Main result | 第59-66页 |
| 3.2 Proof of the Refined Large Deviation Theorem | 第66-71页 |
| 3.3 Appendix 1: Proof of Theorem 3.2 | 第71-77页 |
| 3.4 Appendix 2: Proof of Lemma 3.7,3.8 | 第77-81页 |
| Chapter 4 Mixed spectral types for the one frequency discrete quasi-periodicSchrodinger operator | 第81-87页 |
| 4.1 Introduction | 第81-83页 |
| 4.2 Singular spectrum in the positive Lyapunov exponent region | 第83-84页 |
| 4.3 Absolutely continuous spectrum near the bottom | 第84-87页 |
| Bibliography | 第87-93页 |
