Quantitative Derivation of the Two-Component Gross--Pitaevskii Equation in the Hard-Core Limit with Uniform-in-Time Convergence Rate

Abstract: We derive the time-dependent two-component Gross--Pitaevskii (GP) equation as an effective description of the dynamics of a dilute two-component Bose gas near its ground state, which exhibits a two-component Bose-Einstein condensate, in the GP limit. Our main result establishes a uniform-in-time bound on the convergence rate between the many-body dynamics and the effective description, explicitly quantified in terms of the particle number $N$, and also implies a uniform-in-time bound for the one-component case. This improves upon the works of Michelangeli and Olgliati [77, 89] by providing a sharper, $N$-dependent, time-independent convergence rate. Our approach further extends the framework of Benedikter, de Oliveira, and Schlein [10] to the multi-component Bose gas in the hard-core limit setting. More specifically, we develop the necessary Bogoliubov theory to analyze the dynamics of multi-component Bose gases in the GP regime.