Absence of absolutely continuous spectrum for generic quasi-periodic Schrödinger operators on the real line

We show that a generic quasi-periodic Schrödinger operator in L 2 (ℝ) has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrödinger operat...

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Bibliographic Details
Published inIsrael journal of mathematics Vol. 247; no. 2; pp. 783 - 796
Main Authors Damanik, David, Lenz, Daniel
Format Journal Article
LanguageEnglish
Published Jerusalem The Hebrew University Magnes Press 01.04.2022
Springer Nature B.V
Subjects
Algebra
Analysis
Applications of Mathematics
Continuity (mathematics)
Group Theory and Generalizations
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Operators (mathematics)
Sampling
Theoretical
Toruses
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Summary:We show that a generic quasi-periodic Schrödinger operator in L 2 (ℝ) has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrödinger operator with the resulting potential has empty absolutely continuous spectrum.
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-021-2280-4