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Softcover ISBN:  9781470435431 
Product Code:  MEMO/258/1239 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
eBook ISBN:  9781470450694 
Product Code:  MEMO/258/1239.E 
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MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
Softcover ISBN:  9781470435431 
eBook ISBN:  9781470450694 
Product Code:  MEMO/258/1239.B 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 258; 2019; 139 ppMSC: Primary 35; 81; Secondary 37; 47;
The authors consider a Schrödinger operator \(H=\Delta +V(\vec x)\) in dimension two with a quasiperiodic 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 selfcontained and can be read without familiarity with the previous paper. 
Table of Contents

Chapters

1. Introduction

2. Preliminary Remarks

3. Step I

4. Step II

5. Step III

6. STEP IV

7. Induction

8. Isoenergetic Sets. Generalized Eigenfunctions of $H$

9. Proof of Absolute Continuity of the Spectrum

10. Appendices

11. List of main notations


Additional Material

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The authors consider a Schrödinger operator \(H=\Delta +V(\vec x)\) in dimension two with a quasiperiodic 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 selfcontained and can be read without familiarity with the previous paper.

Chapters

1. Introduction

2. Preliminary Remarks

3. Step I

4. Step II

5. Step III

6. STEP IV

7. Induction

8. Isoenergetic Sets. Generalized Eigenfunctions of $H$

9. Proof of Absolute Continuity of the Spectrum

10. Appendices

11. List of main notations