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Multiscale analysis in momentum space for quasi-periodic potential in dimension two

### Abstract

We consider a polyharmonic operator in dimension two with l ⩾ 2, l being an integer, and a quasi-periodic potential . We 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 at the high energy region. Second, the isoenergetic curves in the space of momenta corresponding to these eigenfunctions have a 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.

© 2013 AIP Publishing LLC

Received 07 February 2013
Accepted 04 June 2013
Published online 11 July 2013

Acknowledgments:
The authors are very grateful to Professor Parnovski for useful discussions and to Professor Young-Ran Lee for allowing us to use Figures 1–4 from Refs. 29–31. Research partially supported by U.S. National Science Foundation (NSF) Grant No. DMS 1201048 (Y.K.).

Article outline:

I. INTRODUCTION
II. PRELIMINARY REMARKS
III. STEP I
A. Operator *H* ^{(1)}
B. Perturbation formulas
C. Geometric considerations
D. Isoenergetic surface for operator *H* ^{(1)}
E. Preparation for step II: Construction of the second nonresonant set
1. Model operator for step II
2. Estimates for the resolvent of the model operator
3. Resonant and nonresonant sets for step II
IV. STEP II
A. Operator *H* ^{(2)}: Perturbation formulas
B. Isoenergetic surface for operator *H* ^{(2)}
C. Preparation for step III: Geometric part. Properties of the quasiperiodic lattice
1. General lemmas
2. Lattice points in the nonresonant set
3. Lattice points in the resonant set
D. Preparation for step III: Analytic part
1. Model operator for step III
2. Resonant and nonresonant sets for step III
V. STEP III
A. Operator *H* ^{(3)}: Perturbation formulas
B. Isoenergetic surface for operator *H* ^{(3)}
C. Preparation for step IV
1. Properties of the quasiperiodic lattice: Continuation
2. Model operator for step IV
3. Resonant and nonresonant sets for step 4
VI. STEP IV
A. Operator *H* ^{(4)}: Perturbation formulas
B. Isoenergetic surface for operator *H* ^{(4)}
VII. INDUCTION
A. Inductive formulas for *r* _{ n }
B. Preparation for step *n* + 1, *n* ⩾ 4
1. Properties of the quasiperiodic lattice: Induction
2. Model operator for step *n* + 1
3. Resonant and nonresonant sets for step *n* + 1
C. Operator *H* ^{(n + 1)}: Perturbation formulas
D. Isoenergetic surface for operator *H* ^{(n + 1)}
VIII. ISOENERGETIC SETS: GENERALIZED EIGENFUNCTIONS OF *H*
A. Construction of limit-isoenergetic set
B. Generalized eigenfunctions of H
IX. PROOF OF ABSOLUTE CONTINUITY OF THE SPECTRUM
A. Operators ,
B. Sets and
C. Projections
D. Proof of absolute continuity

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2013-07-11

2016-08-29

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