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

Marco Falconi, Nikolai Leopold, David Mitrouskas, Sören Petrat

Abstract

We study the time evolution of the Nelson model in a mean-field limit in which N non-relativistic 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 first- and second-order approximation to the one-particle reduced density matrices of the particles and the quantum field, respectively.

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 non-relativistic 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 first- and second-order approximation to the one-particle reduced density matrices of the particles and the quantum field, respectively.

Paper Structure

This paper contains 14 sections, 10 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and notations
  3. Main Results
  4. Bogoliubov Theory
  5. Higher-Order Corrections
  6. Application of Main Results
  7. Explicit action of the Bogoliubov time evolution
  8. Equations for the reduced density and next-order corrections
  9. Well-posedness of the Bogoliubov Equation
  10. Proofs
  11. Preliminary Lemmas
  12. Propagation of Moments of Number Operators
  13. Proof of the Theorems
  14. More details on the excitation Fock spaces

Key Result

Lemma \oldthetheorem

Let $(u_0, \alpha_0) \in H^2(\mathbb{R}^3) \times L_1^2(\mathbb{R}^3)$. Then there is a continuous map $t\mapsto (u(t), \alpha(t))$ from $\mathbb{R}$ to $H^2(\mathbb{R}^3) \oplus L_1^2(\mathbb{R}^3)$ that satisfies varphi_eq2-alpha_eq2 with initial condition $(u(t), \alpha(t)) |_{t=0} = (u_0, \alpha

Theorems & Definitions (21)

  • Lemma \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 11 more