### Abstract

We consider the Schrödinger equation with a random potential of the form
V(x) = w^2(x)/4-w'(x)/2
where w is a Lévy noise. We focus on the problem of computing the so-called complex Lyapunov exponent
Ω = γ - iπN
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.

Translated title of the contribution | Supersymmetric Quantum Mechanics with Lévy Disorder in One Dimension |
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Original language | English |

Pages (from-to) | 1291 - 1323 |

Number of pages | 33 |

Journal | Journal of Statistical Physics |

Volume | 145 |

Issue number | 5 |

DOIs | |

Publication status | Published - Dec 2011 |

### Bibliographical note

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## Cite this

Comtet, A., Texier, C., & Tourigny, YJM. (2011). Supersymmetric Quantum Mechanics with Lévy Disorder in One Dimension.

*Journal of Statistical Physics*,*145*(5), 1291 - 1323. https://doi.org/10.1007/s10955-011-0351-3