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Universal Low-energy Behavior in a Quantum Lorentz Gas with Gross-Pitaevskii Potentials

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Abstract

We consider a quantum particle interacting with N obstacles, whose positions are independently chosen according to a given probability density, through a two-body potential of the form N2V (Nx) (Gross-Pitaevskii potential). We show convergence of the N dependent one-particle Hamiltonian to a limiting Hamiltonian where the quantum particle experiences an effective potential depending only on the scattering length of the unscaled potential and the density of the obstacles. In this sense our Lorentz gas model exhibits a universal behavior for N large. Moreover we explicitely characterize the fluctuations around the limit operator. Our model can be considered as a simplified model for scattering of slow neutrons from condensed matter.

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Acknowledgments

The authors gratefully acknowledge the support from the GNFM Gruppo Nazionale per la Fisica Matematica - INDAM.

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Authors and Affiliations

  1. Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057, Zurich, Switzerland

    Giulia Basti

  2. Gran Sasso Science Institute, Viale Francesco Crispi 7, 67100, L’Aquila, Italy

    Serena Cenatiempo

  3. Dipartimento di Matematica G. Castelnuovo, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy

    Alessandro Teta

Authors
  1. Giulia Basti
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  2. Serena Cenatiempo
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  3. Alessandro Teta
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Correspondence to Giulia Basti.

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Basti, G., Cenatiempo, S. & Teta, A. Universal Low-energy Behavior in a Quantum Lorentz Gas with Gross-Pitaevskii Potentials. Math Phys Anal Geom 21, 11 (2018). https://doi.org/10.1007/s11040-018-9268-2

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  • DOI: https://doi.org/10.1007/s11040-018-9268-2

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