Pair excitations and the mean field approximation of interacting Bosons, I

TL;DR

This paper extends the mean field approximation for interacting Bosons by deriving linear equations for pair excitations under stronger interactions, providing estimates that compare exact and approximate dynamics.

Contribution

It introduces a new derivation of pair excitation evolution equations for stronger interactions, with particle-number independent estimates.

Findings

Derived linear equations for pair excitations.

Provided estimates independent of particle number.

Compared exact and approximate dynamics effectively.

Abstract

In our previous work \cite{GMM1},\cite{GMM2} we introduced a correction to the mean field approximation of interacting Bosons. This correction describes the evolution of pairs of particles that leave the condensate and subsequently evolve on a background formed by the condensate. In \cite{GMM2} we carried out the analysis assuming that the interactions are independent of the number of particles $N$. Here we consider the case of stronger interactions. We offer a new transparent derivation for the evolution of pair excitations. Indeed, we obtain a pair of linear equations describing their evolution. Furthermore, we obtain apriory estimates independent of the number of particles and use these to compare the exact with the approximate dynamics.