Must the Spectrum of a Random Schrödinger Operator Contain an Interval?

Damanik, David; Gorodetski, Anton

doi:10.48550/arxiv.2112.02445

2021

DOI: 10.48550/arxiv.2112.02445

|View full text |Cite

Preprint

Must the Spectrum of a Random Schrödinger Operator Contain an Interval?

David Damanik¹,

Anton Gorodetski²

Abstract: We consider Schrödinger operators in 2 (Z) whose potentials are given by independent (not necessarily identically distributed) random variables. We ask whether it is true that almost surely its spectrum contains an interval. We provide an affirmative answer in the case of random potentials given by a sum of a perturbatively small quasi-periodic potential with analytic sampling function and Diophantine frequency vector and a term of Anderson type, given by independent identically distributed random variables. T… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2022

Publication Types

Select...

Other1

Relationship

Self Cite1

Independent0

Authors

Journals

Cited by 1 publication

(1 citation statement)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another class of examples in the closely related continuum setting is given by sums of two periodic potentials with incommensurate frequencies, which provide the simplest examples of quasi-periodic potentials; compare [38,41,68] for an incomplete list. This is but a partial list of potential settings; other recent papers consider more general additive perturbations of random [14,24] and quasiperiodic [74] potentials.…”

mentioning

confidence: 99%

Spectral Characteristics of Schrödinger Operators Generated by Product Systems

Damanik¹,

Fillman²,

Gohlke³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

Motivated by the question of what spectral properties of dynamically defined Schrödinger operators may be preserved under periodic perturbations, we study ergodic Schrödinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational rotation of the circle. The scenario given by a periodic background potential corresponds to a separable structure in which the sampling function is the sum of two pieces, each of which depends only on a single factor of the product system. However, in each case that we study, our methods apply more generally to sampling functions that allow nontrivial dependencies between the product factors.In the case in which one factor is a Boshernitzan subshift, we prove that either the resulting operators are periodic or the resulting spectra must be Cantor sets. The main ingredient is a suitable stability result for Boshernitzan's criterion under taking products. We also discuss the stability of purely singular continuous spectrum, which, given the zero-measure spectrum result, amounts to stability results for eigenvalue exclusion. In particular, we examine situations in which the existing criteria for the exclusion of eigenvalues are stable under periodic perturbations. As a highlight of this, we show that any simple Toeplitz subshift over a binary alphabet exhibits uniform absence of eigenvalues on the hull for any periodic perturbation whose period is commensurate with the coding sequence. This is new, even in the case in which the periodic background vanishes entirely. In the case of a full shift, we give an effective criterion to compute exactly the spectrum of a random Anderson model perturbed by a potential of period two, and we further show that the naive generalization of this criterion does not hold for period three. Next, we consider quasi-periodic potentials with potentials generated by trigonometric polynomials with periodic background. We show that the quasiperiodic cocycle induced by passing to blocks of period length is subcritical when the coupling constant is small and supercritical when the coupling constant is large. Thus, the spectral type is absolutely continuous for small coupling and pure point (for a.e. frequency and phase) when the coupling is large. Contents 2.3. Cocycles and Hyperbolicity 8 3. Periodic and Subshift 9 3.1. Zero-Measure Cantor Spectrum 9 3.2. Exclusion of Eigenvalues 19 3.3. Periodic and Bernoulli 30 4. Periodic and One-Frequency Quasi-Periodic 34 4.1. Generalities 34 4.2. Consequences of Global Theory 35 Appendix A. Ergodic Measures on Accelerated Systems 39 Appendix B. Strict Ergodicity of Product Systems 41 References 43

show abstract

mentioning

confidence: 99%

Spectral Characteristics of Schrödinger Operators Generated by Product Systems

Damanik¹,

Fillman²,

Gohlke³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Must the Spectrum of a Random Schrödinger Operator Contain an Interval?

Cited by 1 publication

References 33 publications

Spectral Characteristics of Schrödinger Operators Generated by Product Systems

Spectral Characteristics of Schrödinger Operators Generated by Product Systems

Contact Info

Product

Resources

About