Extended States for the Schrödinger Operator with Quasi-Periodic Potential in Dimension Two.

Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Preliminary Remarks -- Chapter 3. Step I -- 3.1. The Operator ⁽¹⁾ -- 3.2. Perturbation Formulas -- 3.3. Geometric Considerations -- 3.4. Isoenergetic Surface for the Operator ⁽¹⁾ -- 3.5. Preparation for Step II. Construction of the Second Nonresonant Set -- Chapter 4. Step II -- 4.1. The Operator ⁽²⁾. Perturbation Formulas -- 4.2. Isoenergetic Surface for the Operator ⁽²⁾ -- 4.3. Preparation for Step III - Geometric Part. Properties of the Quasiperiodic Lattice -- 4.4. Preparation for Step III - Analytic Part -- Chapter 5. Step III -- 5.1. The Operator ⁽³⁾. Perturbation Formulas -- 5.2. Isoenergetic Surface for the Operator ⁽³⁾ -- 5.3. Preparation for Step IV -- Chapter 6. STEP IV -- 6.1. The Operator ⁽⁴⁾. Perturbation Formulas -- 6.2. Isoenergetic Surface for the Operator ⁽⁴⁾ -- Chapter 7. Induction -- 7.1. Inductive formulas for _{ } -- 7.2. Preparation for Step +1, ≥4 -- 7.3. The Operator ⁽ⁿ⁺¹⁾. Perturbation Formulas -- 7.4. Isoenergetic Surface for the Operator ⁽ⁿ⁺¹⁾ -- Chapter 8. Isoenergetic Sets. Generalized Eigenfunctions of -- 8.1. Construction of the Limit-Isoenergetic Set -- 8.2. Generalized Eigenfunctions of -- Chapter 9. Proof of Absolute Continuity of the Spectrum -- 9.1. The Operators _{ }( _{ }'), _{ }'⊂ _{ } -- 9.2. Sets _{∞} and _{∞, } -- 9.3. Projections ( _{∞, }) -- 9.4. Proof of Absolute Continuity -- Chapter 10. Appendices -- 10.1. Appendix 1. Proof of Lemma 3.12 -- 10.2. Appendix 2. Proof of Lemma 3.13 -- 10.3. Appendix 3 -- 10.4. Appendix 4 -- 10.5. Appendix 5 -- 10.6. Appendix 6 -- 10.7. Appendix 7 -- 10.8. Appendix 8. An Application of Bezout's Theorem -- 10.9. Appendix 9. On the Proof of Geometric Lemmas Allowing to Deal with Clusters instead of Boxes -- 10.10. Appendix 10 -- Chapter 11. List of main notations.

Bibliography -- Back Cover.

The authors consider a Schrödinger operator H=-\Delta +V(\vec x) in dimension two with a quasi-periodic potential V(\vec x). They prove that the absolutely continuous spectrum of H contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves e^i\langle \vec \varkappa ,\vec x\rangle in the high energy region. Second, the isoenergetic curves in the space of momenta \vec \varkappa corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results. The result is based on a previous paper on the quasiperiodic polyharmonic operator (-\Delta )^l+V(\vec x), l>1. Here the authors address technical complications arising in the case l=1. However, this text is self-contained and can be read without familiarity with the previous paper.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.