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 [Formula: see text] nonrelativistic bosons weakly couple (with respect to 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 partial differential equations describing the time evolution of the first- and second-order approximations to the one-particle reduced density matrices of the particles and the quantum field, respectively.

Asymptotic expansion of low-energy excitations for weakly interacting bosons

“…Our article is an application of that expansion for the ground state energy. Note that the work [6] was in turn inspired by an analogous result for the dynamics [5]; see also the follow-up work [7]. Let us refer to [2] and [3] for reviews of both results, and note that in [4] the results from [6] are applied to derive an Edgeworth expansion for the fluctuations of bounded one-body operators with respect to the ground state and low-energy excited states of the weakly interacting Bose gas.…”

“…with initial datum (ϕ 0 , α 0 ), where |ϕ(t)| 2 is the Fourier transform of |ϕ(t)| 2 . In [21] it was shown that (for suitably chosen initial states) the time evolution of the regularized Nelson model satisfies an asymptotic expansion in the spirit of Theorem 4. The main difference to the previous subsection, where N bosons interact via pair potentials, is that the system consists now of two types of particles.…”

“…It is our pleasure to thank Marco Falconi, Nataša Pavlović, Peter Pickl, Robert Seiringer and Avy Soffer for the collaboration on the works [11,13,14,21,33,39]. L.B.…”

“…In Section 3, we review the Edgeworth expansion from [12] and the binding energy expansion from [10]. Finally, in Section 4, we review the dynamical results from [13] and [21].…”