Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential

Abstract

Consider a system of N N bosons in three dimensions interacting via a repulsive short range pair potential N 2 V ( N ( x i − x j ) ) N^2V(N(x_i-x_j)) , where x = ( x 1 , … , x N ) \mathbf {x}=(x_1, \ldots , x_N) denotes the positions of the particles. Let H N H_N denote the Hamiltonian of the system and let ψ N , t \psi _{N,t} be the solution to the Schrödinger equation. Suppose that the initial data ψ N , 0 \psi _{N,0} satisfies the energy condition \[ ⟨ ψ N , 0 , H N ψ N , 0 ⟩ ≤ C N \langle \psi _{N,0}, H_N \psi _{N,0} \rangle \leq C N \] and that the one-particle density matrix converges to a projection as N → ∞ N \to \infty . Then, we prove that the k k -particle density matrices of ψ N , t \psi _{N,t} factorize in the limit N → ∞ N \to \infty . Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic nonlinear Schrödinger equation with the coupling constant proportional to the scattering length of the potential V V . In a recent paper, we proved the same statement under the condition that the interaction potential V V is sufficiently small. In the present work we develop a new approach that requires no restriction on the size of the potential.

Quantitative Derivation of the Gross‐Pitaevskii Equation

“…In fact, the following result was established in [6,7,8,9,10], and, in a slightly different form, in [21]. Consider a family ψ N ∈ L 2 s (R 3N ) with bounded energy per particle…”

“…Then, the solution ψ N,t of the Schrödinger equation (8) still exhibits complete Bose-Einstein condensation, in the sense that the reduced one-particle density γ N,t → |ϕ t ϕ t | (10) as N → ∞, where ϕ t is the solution of the time-dependent Gross-Pitaevskii equation i∂ t ϕ t = −∆ϕ t + 8πa 0 |ϕ t | 2 ϕ t .…”

“…In contrast to [7,8,9,10], the approach presented in this paper can be extended with no additional complication to Hamilton operators with external potential. This remark is important to describe experiments where the evolution of the condensate is observed after tuning the magnetic traps, rather than switching them off.…”

Derivation of Hartreeʼs theory for generic mean-field Bose systems

“…There are many works on the related (but still different) derivation of the time-dependent Hartree theory from the time-dependent Schrödinger equation associated with the Hamiltonian H N , see for instance [29,25,63,7,20,21,3,22,24,54,34,51]. In this case one starts close to a Hartree state at time zero, and then proves that the Schrödinger flow stays close to the corresponding trajectory of the Hartree state.…”

Randomization and the Gross–Pitaevskii Hierarchy

“…In a sequence of monumental works, Erdos, Schlein and Yau [77][78][79][80] gave a rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on R 3 . In the aforementioned works, a significant step was to check the uniqueness of solutions to (1).…”