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Optimal rate of condensation for trapped bosons in the Gross–Pitaevskii regime

Phan Thành Nam, Marcin Napiórkowski, Julien Ricaud and Arnaud Triay

Vol. 15 (2022), No. 6, 1585–1616
Abstract

We study the Bose–Einstein condensates of trapped Bose gases in the Gross–Pitaevskii regime. We show that the ground state energy and ground states of the many-body quantum system are correctly described by the Gross–Pitaevskii equation in the large particle number limit, and provide the optimal convergence rate. Our work extends the previous results of Lieb, Seiringer and Yngvason on the leading-order convergence, and of Boccato, Brennecke, Cenatiempo and Schlein on the homogeneous gas. Our method relies on the idea of “completing the square”, inspired by recent works of Brietzke, Fournais and Solovej on the Lee–Huang–Yang formula, and a general estimate for Bogoliubov quadratic Hamiltonians on Fock space.

Keywords
trapped Bose gases, Gross–Pitaevskii equation, Bose–Einstein condensation, optimal bounds
Mathematical Subject Classification
Primary: 81V70, 81V73
Milestones
Received: 7 July 2020
Revised: 3 December 2020
Accepted: 10 February 2021
Published: 10 November 2022
Authors
Phan Thành Nam
Department of Mathematics
Ludwig Maximilian University of Munich
Munich
Germany
Munich Center for Quantum Science and Technology
Munich
Germany
Marcin Napiórkowski
Department of Mathematical Methods in Physics
Faculty of Physics
University of Warsaw
Warsaw
Poland
Julien Ricaud
Department of Mathematics
Ludwig Maximilian University of Munich
Munich
Germany
Munich Center for Quantum Science and Technology
Munich
Germany
Arnaud Triay
Department of Mathematics
Ludwig Maximilian University of Munich
Munich
Germany
Munich Center for Quantum Science and Technology
Munich
Germany