Abstract
We consider the homogeneous Bose gas on a unit torus in the mean-field regime when the interaction strength is proportional to the inverse of the particle number. In the limit when the number of particles becomes large, we derive a two-term expansion of the one-body density matrix of the ground state. The proof is based on a cubic correction to Bogoliubov’s approximation of the ground state energy and the ground state.
Highlights
We consider a homogeneous system of N bosons on the unit torus Td, for any dimension d ≥ 1
In Lemma 9 we prove that e( p)a∗pap + R3 p=0 with an error term R3 whose expectation against the ground state is of order O(N −3/2)
We justify the Bose–Einstein condensation by showing that the ground state has a bounded number of excited particles
Summary
We consider a homogeneous system of N bosons on the unit torus Td , for any dimension d ≥ 1. The system is governed by the mean-field Hamiltonian. The kinetic operator − is the usual Laplacian (with periodic boundary conditions). The interaction potential w is a real-valued, even function.
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