[Paper Review] Bose particles in a box II. A convergent expansion of the ground state of the Bogoliubov Hamiltonian in the mean field limiting regime

Alessandro Pizzo|arXiv (Cornell University)|Nov 22, 2015

Cold Atom Physics and Bose-Einstein Condensates17 references22 citations

TL;DR

This paper presents a convergent perturbative expansion for the ground state of the particle-number-conserving Bogoliubov Hamiltonian in an interacting Bose gas within a finite box, under the mean field limiting regime. Using a multi-scale analysis in occupation numbers of particle states, the authors derive a systematic expansion in terms of bare operators that converges as N → ∞, providing a rigorous construction of the ground state up to arbitrary precision.

ABSTRACT

In this paper we consider an interacting Bose gas at zero temperature, in a finite box and in the mean field limiting regime. The N gas particles interact through a pair potential of positive type and with an ultraviolet cut-off. Its (nonzero) Fourier components are sufficiently large with respect to the corresponding kinetic energies of the modes. Using the multi-scale technique in the occupation numbers of particle states introduced in [Pi1], we provide a convergent expansion of the ground state of the particle number preserving Bogoliubov Hamiltonian in terms of the bare operators. In the limit N o \infty the expansion is up to any desired precision.

Motivation & Objective

• To rigorously construct the ground state of an interacting Bose gas in a finite box at zero temperature.
• To extend the multi-scale technique in occupation numbers—previously applied to three-mode systems—to the full Bogoliubov Hamiltonian with many interacting modes.
• To establish a convergent expansion of the ground state vector in the mean field limiting regime, where the coupling constant scales inversely with particle density.
• To provide a non-perturbative, systematic expansion in bare operators that holds up to any desired precision as N → ∞.
• To validate the applicability of Feshbach-Schur flows across multiple mode pairs in a hierarchical, iterative manner.

Proposed method

• Employ a multi-scale analysis in the occupation numbers of single-particle states, building on techniques from the companion paper [Pi1] on three-mode systems.
• Apply iterative Feshbach-Schur flows to decouple interacting mode pairs, treating each pair as a three-mode subsystem.
• Use a complex parameterization of the Hamiltonian flow to control the convergence of the expansion via analyticity and bounds on operator norms.
• Introduce a regularization scheme with an ultraviolet cutoff and control the dependence on the particle number N through rescaling of mode occupation numbers.
• Derive recursive bounds on the derivatives of Green's functions and wave operator components with respect to occupation number shifts.
• Establish uniform estimates on the convergence rate using induction, showing that the derivative of the Green's function grows at most as O(h·g^{i−N+h}/√N) for g ∈ (3,4).

Experimental results

Research questions

• RQ1 Can a convergent expansion of the ground state vector be constructed for the full particle-number-preserving Bogoliubov Hamiltonian in the mean field limit?
• RQ2 How can the multi-scale technique in occupation numbers be generalized from three-mode systems to a full many-mode Hamiltonian?
• RQ3 What are the quantitative bounds on the convergence of the expansion in terms of the particle number N and mode interactions?
• RQ4 How do the Feshbach-Schur flows behave across multiple mode pairs, and can they be controlled uniformly in the large-N limit?
• RQ5 What is the role of the ultraviolet cutoff and the strong interaction assumption in ensuring the convergence of the expansion?

Key findings

• A convergent expansion of the ground state vector is constructed for the Bogoliubov Hamiltonian in the mean field regime, valid up to any desired precision as N → ∞.
• The expansion is expressed in terms of bare creation and annihilation operators, with all corrections systematically resummed via the multi-scale scheme.
• The method ensures uniform convergence through recursive bounds on the derivatives of Green's functions and wave operators with respect to occupation number shifts.
• The bound on the derivative of the Green's function component is shown to be O(h·g^{i−N+h}/√N) for g ∈ (3,4), confirming the exponential decay of higher-order corrections.
• The analysis confirms the validity of the Feshbach-Schur flow approach across multiple mode pairs, enabling a hierarchical, iterative construction of the ground state.
• The convergence is established under the strong interaction assumption, where Fourier components of the potential are sufficiently large compared to kinetic energies.