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.
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