Ground state energy of mixture of Bose gases

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We consider the asymptotic behavior of a system of multi-component trapped bosons, when the total particle number N becomes large. In the dilute regime, when the interaction potentials have the length scale of order O(N-1), we show that the leading order of the ground state energy is captured correctly by the Gross-Pitaevskii energy functional and that the many-body ground state fully condensates on the Gross- Pitaevskii minimizers. In the mean-field regime, when the interaction length scale is O(1), we are able to verify Bogoliubov's approximation and obtain the second order expansion of the ground state energy. While such asymptotic results have several precursors in the literature on one-component condensates, the adaption to the multi-component setting is non-trivial in various respects and the analysis will be presented in details
Contents: Introduction. 2. Main results. 3. Proof of Theorem 2.1. 3.1. GP minimiser. 3.2. Energy upper bound. 3.3. Dyson Lemma 16 3.4. Energy lower bound 3.5. Convergence of density matrices 4. Proof of Theorem 2.2. 4.1. Leading order and Hartree theory. 4.2. Bogoliubov Hamiltonian 4.3. Estimate in the truncated two-component Fock space. 4.4. Localization in Fock space. 4.5. Validity of Bogoliubov correction. Appendix A. Quantum de Finetti Theorem 3.4. References