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The Gross–Pitaevskii Hierarchy on General Rectangular Tori

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Abstract

In this work, we study the Gross–Pitaevskii hierarchy on general—rational and irrational—rectangular tori of dimensions two and three. This is a system of infinitely many linear partial differential equations which arises in the rigorous derivation of the nonlinear Schrödinger equation. We prove a conditional uniqueness result for the hierarchy. In two dimensions, this result allows us to obtain a rigorous derivation of the defocusing cubic nonlinear Schrödinger equation from the dynamics of many-body quantum systems. On irrational tori, this question was posed as an open problem in the previous work of Kirkpatrick, Schlein, and Staffilani.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501, Bielefeld, Germany

    Sebastian Herr

  2. Departement Mathematik, Eidgenössische Technische Hochschule Zürich, Rämistrasse 101, 8092, Zurich, Switzerland

    Vedran Sohinger

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  1. Sebastian Herr
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Correspondence to Sebastian Herr.

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Communicated by G. Friesecke

This work has been partially supported by the German Research Foundation, CRC 701.

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Herr, S., Sohinger, V. The Gross–Pitaevskii Hierarchy on General Rectangular Tori. Arch Rational Mech Anal 220, 1119–1158 (2016). https://doi.org/10.1007/s00205-015-0950-2

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