Bogoliubov Dynamics and Higher-order Corrections for the Regularized Nelson Model

Abstract

We study the time evolution of the Nelson model in a mean-field limit in which N nonrelativistic bosons weakly couple (w.r.t. the particle number) to a positive or zero mass quantized scalar field. Our main result is the derivation of the Bogoliubov dynamics and higher-order corrections. More precisely, we prove the convergence of the approximate wave function to the many-body wave function in norm, with a convergence rate proportional to the number of corrections taken into account in the approximation. We prove an analogous result for the unitary propagator. As an application, we derive a simple system of PDEs describing the time evolution of the firstand second-order approximation to the one-particle reduced density matrices of the particles and the quantum field, respectively. MSC class: 35Q40, 35Q55, 81Q05, 81T10, 81V73, 82C10