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arXiv:1511.07022

Bose particles in a box I. A convergent expansion of the ground state of a three-modes Bogoliubov Hamiltonian

Alessandro Pizzo

Published Nov 22, 2015• 14 citations
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Abstract

In this paper we introduce a novel multi-scale technique to study many-body quantum systems where the total number of particles is kept fixed. The method is based on Feshbach map and the scales are represented by occupation numbers of particle states. Here, we consider a three-modes (including the zero mode) Bogoliubov Hamiltonian for a sufficiently small ratio between the kinetic energy and the Fourier component of the (positive type) potential corresponding to the two nonzero modes. For any space dimension d\geq 1 and in the mean field limiting regime (i.e., at fixed box volume |Λ| and for a number of particles, N, sufficiently large) this method provides the construction of the ground state and its expansion in terms of the bare operators that in the limit N \to \infty is up to any desired precision. In space dimension d \geq 3 the method provides similar results for an arbitrarily large (finite) box and a large but fixed particle density ρ, i.e.,ρis independent of the size of the box.

DOI
10.48550/arxiv.1511.07022

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In this paper we introduce a novel multi-scale technique to study many-body quantum systems where the total number of particles is kept fixed. The method is based on Feshbach map and the scales are represented by occupation numbers of particle states. Here, we consider a three-modes (including the zero mode) Bogoliubov Hamiltonian for a sufficiently small ratio between the kinetic energy and the Fourier component of the (positive type) potential corresponding to the two nonzero modes. For any space dimension d\geq 1 and in the mean field limiting regime (i.e., at fixed box volume |Λ| and for a number of particles, N, sufficiently large) this method provides the construction of the ground state and its expansion in terms of the bare operators that in the limit N \to \infty is up to any desired precision. In space dimension d \geq 3 the method provides similar results for an arbitrarily large (finite) box and a large but fixed particle density ρ, i.e.,ρis independent of the size of the box.
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 \to \infty the expansion is up to any desired precision.
In this paper we consider an interacting Bose gas at zero temperature, constrained to 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. The (nonzero) Fourier components of the potential are assumed to be sufficiently large with respect to the corresponding kinetic energies of the modes like in the companion papers [Pi1]-[Pi2]. Using the multi-scale technique in the occupation numbers of particle states introduced in [Pi1]-[Pi2], we provide a convergent expansion of the ground state of the Hamiltonian in terms of the bare operators. In the limit N \to \infty the expansion is up to any desired precision.
We study the ground state for many interacting bosons in a double-well potential, in a joint limit where the particle number and the distance between the potential wells both go to infinity.Two single-particle orbitals (one for each well) are macroscopically occupied, and we are concerned with deriving the corresponding effective Bose-Hubbard Hamiltonian.We prove an energy expansion, including the two-mode Bose-Hubbard energy and two independent Bogoliubov corrections (one for each potential well), and a variance bound for the number of particles falling inside each potential well.The latter is a signature of a correlated ground state in that it violates the central limit theorem.1. Introduction 1885 2. Main statements 1889 3. Mapping to the space of excitations 1897 4. Bounds on the two-mode Hamiltonian 1903 5. Derivation of the Bogoliubov Hamiltonian and reduction to right and left modes 1910 6.A priori estimates on the ground state of H N 1926 7. Shifted Hamiltonians and lower bound 1932 8. Proof of the main results 1939 Appendix A. The one-body Hartree problem 1949 Appendix B. Estimates and identities in the two-mode space 1951
We consider the asymptotic behavior of a system of multi-component trapped bosons, when the total particle number [Formula: see text] becomes large. In the dilute regime, when the interaction potentials have the length scale of order [Formula: see text], we show that the leading order of the ground state energy is captured correctly by the Gross–Pitaevskii energy functional and that the many-body ground state fully condensates on the Gross–Pitaevskii minimizers. In the mean-field regime, when the interaction length scale is [Formula: see text], we are able to verify Bogoliubov’s approximation and obtain the second order expansion of the ground state energy. While such asymptotic results have several precursors in the literature on one-component condensates, the adaptation to the multi-component setting is non-trivial in various respects and the analysis will be presented in detail.