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Ballistic Transport for the Schrödinger Operator with Limit-Periodic or Quasi-Periodic Potential in Dimension Two

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Abstract

We prove the existence of ballistic transport for the Schrödinger operator with limit-periodic or quasi-periodic potential in dimension two. This is done under certain regularity assumptions on the potential which have been used in prior work to establish the existence of an absolutely continuous component and other spectral properties. The latter include detailed information on the structure of generalized eigenvalues and eigenfunctions. These allow one to establish the crucial ballistic lower bound through integration by parts on an appropriate extension of a Cantor set in momentum space, as well as through stationary phase arguments.

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Authors and Affiliations

  1. Department of Mathematics, Campbell Hall, University of Alabama at Birmingham, 1300 University Boulevard, Birmingham, AL, 35294, USA

    Yulia Karpeshina, Roman Shterenberg & Günter Stolz

  2. Department of Mathematics, Sogang University, 5 Baekbeom-ro, Mapo-gu, Seoul, 121-742, South Korea

    Young-Ran Lee

Authors
  1. Yulia Karpeshina
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  2. Young-Ran Lee
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  3. Roman Shterenberg
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  4. Günter Stolz
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Correspondence to Yulia Karpeshina.

Additional information

Communicated by Y. Karpeshina

Supported in part by NSF-Grants DMS-1201048 (Y.K.) and DMS-1069320 (G.S.), National Research Foundation of Korea (NRF) grants funded by the Korea Government (MSIP) No. 2011-0013073 and (MOE) No. 2014R1A1A2058848 (Y.-R.L.) and Simons Foundation Grant No. 312879 (R.S.).

The authors are thankful to the Isaac Newton Institute for Mathematical Sciences (Cambridge University, UK) for support and hospitality during the program Periodic and Ergodic Spectral Problems where work on this paper was undertaken.

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Karpeshina, Y., Lee, YR., Shterenberg, R. et al. Ballistic Transport for the Schrödinger Operator with Limit-Periodic or Quasi-Periodic Potential in Dimension Two. Commun. Math. Phys. 354, 85–113 (2017). https://doi.org/10.1007/s00220-017-2911-0

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  • DOI: https://doi.org/10.1007/s00220-017-2911-0

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