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Supersymmetric Quantum Mechanics with Lévy Disorder in One Dimension

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Abstract

We consider the Schrödinger equation with a random potential of the form

where w is a Lévy noise. We focus on the problem of computing the so-called complex Lyapunov exponent

where N is the integrated density of states of the system, and γ is the Lyapunov exponent. In the case where the Lévy process is non-decreasing, we show that the calculation of Ω reduces to a Stieltjes moment problem, we ascertain the low-energy behaviour of the density of states in some generality, and relate it to the distributional properties of the Lévy process. We review the known solvable cases—where Ω can be expressed in terms of special functions—and discover a new one.

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Authors and Affiliations

  1. UPMC Univ. Paris 6, 75005, Paris, France

    Alain Comtet

  2. Univ. Paris Sud, CNRS, LPTMS, UMR 8626, Orsay, 91405, France

    Alain Comtet & Christophe Texier

  3. LPS, UMR 8502, Orsay, 91405, France

    Christophe Texier

  4. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    Yves Tourigny

Authors
  1. Alain Comtet
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  2. Christophe Texier
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  3. Yves Tourigny
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Corresponding author

Correspondence to Christophe Texier.

Additional information

This work was supported by “Triangle de la Physique”.

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Comtet, A., Texier, C. & Tourigny, Y. Supersymmetric Quantum Mechanics with Lévy Disorder in One Dimension. J Stat Phys 145, 1291–1323 (2011). https://doi.org/10.1007/s10955-011-0351-3

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  • DOI: https://doi.org/10.1007/s10955-011-0351-3

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