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Inverse Problem for the Discrete 1D Schrödinger Operator with Small Periodic Potentials

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Abstract

Consider the discrete 1D Schrödinger operator on ℤ with an odd 2k periodic potential q. For small potentials we show that the mapping: q→ heights of vertical slits on the quasi-momentum domain (similar to the Marchenko-Ostrovski maping for the Hill operator) is a local isomorphism and the isospectral set consists of 2k distinct potentials. Finally, the asymptotics of the spectrum are determined as q→0.

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Authors and Affiliations

  1. Institut für Mathematik, Humboldt Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

    Evgeny Korotyaev

  2. Faculty of Math. and Mech, St-Petersburg State University,  

    Anton Kutsenko

Authors
  1. Evgeny Korotyaev
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  2. Anton Kutsenko
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Correspondence to Evgeny Korotyaev.

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Communicated by B. Simon

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Korotyaev, E., Kutsenko, A. Inverse Problem for the Discrete 1D Schrödinger Operator with Small Periodic Potentials. Commun. Math. Phys. 261, 673–692 (2006). https://doi.org/10.1007/s00220-005-1429-z

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  • DOI: https://doi.org/10.1007/s00220-005-1429-z

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