Ground states of large bosonic systems : the Gross–Pitaevskii limit revisited

Abstract

Abstract. We study the ground state of a dilute Bose gas in a scaling limit where the Gross-Pitaevskii functional emerges. This is a repulsive non-linear Schrödinger functional whose quartic term is proportional to the scattering length of the interparticle interaction potential. We propose a new derivation of this limit problem, with a method that bypasses some of the technical difficulties that previous derivations had to face. The new method is based on a combination of Dyson's lemma, the quantum de Finetti theorem and a second moment estimate for ground states of the effective Dyson Hamiltonian. It applies equally well to the case where magnetic fields or rotation are present.

Bogoliubov theory for dilute Bose gases: The Gross-Pitaevskii regime

“…It follows from the results of [35,32,33,40] that the ground state energy E N of the Gross-Pitaevskii Hamiltonian…”

Strong Kac’s chaos in the mean-field Bose–Einstein Condensation

“…It is known as Gross-Pitaevskii scaling limit and it involves the scattering length of the interaction potential. The convergence of the ground state energy in this setting is difficult and has been provided by [34,35,43].…”

“…There are many results and some quantitative estimates of the convergence rate for small values of β (see [33,50] and references therein). We face the general mean-field convergence problem by using the hard results for the case β = 1, known as Gross-Pitaevskii scaling limit, obtained in [34,35] and, recently, in [43]. We prove the convergence of the one-particle ground-state energy to the ground-state energy of the non-linear Schroedinger functional for the case of purely repulsive interacting potential (Theorem 2.2).…”

Norm Approximation for Many-Body Quantum Dynamics and Bogoliubov Theory

“…Note that when β = 1 (the Gross-Pitaevskii regime), the strong correlations between particles require a subtle correction: the nonlinear term w N * |u(t)| 2 in Hartree equation (3) has to be replaced by 8πa|u(t)| 2 with a being the scattering length of w. This has been justified rigorously in the context of the Bose-Einstein condensation (11); see [34,33,43] for the ground state problem and [18,17,8,45] for the dynamical problem. The norm approximation is completely open.…”