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

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Abstract

We show that a generic quasi-periodic Schrödinger operator in L2(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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Rice University, Houston, TX, 77005, USA

    David Damanik

  2. Institute for Mathematics, Friedrich-Schiller University, 07743, Jena, Germany

    Daniel Lenz

Authors
  1. David Damanik
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  2. Daniel Lenz
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Corresponding author

Correspondence to Daniel Lenz.

Additional information

was supported in part by NSF grant DMS-1700131.

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Damanik, D., Lenz, D. Absence of absolutely continuous spectrum for generic quasi-periodic Schrödinger operators on the real line. Isr. J. Math. 247, 783–796 (2022). https://doi.org/10.1007/s11856-021-2280-4

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  • DOI: https://doi.org/10.1007/s11856-021-2280-4

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