Abstract
In this paper, we describe a multi-scale technique introduced in Pizzo (Bose particles in a box I. A convergent expansion of the ground state of a three-modes Bogoliubov Hamiltonian, [32]) to study many-body quantum systems. The method is based on the Feshbach–Schur map and the scales are represented by occupation numbers of particle states. The main purpose of this method is to implement singular perturbation theory to deal with large field problems. A simple model to apply our method is the three-modes (including the zero mode) Bogoliubov Hamiltonian that here we consider for a sufficiently small ratio between the kinetic energy and the Fourier component of the (positive type) potential corresponding to the two nonzero modes. In space dimension \(d\ge 3\), for an arbitrarily large box and at fixed, large particle density \(\rho \) (i.e., \(\rho \) is independent of the size of the box), this method provides the construction of the ground state vector of the system and its expansion, up to any desired precision, in terms of the bare operators and the ground state energy. In the mean field limiting regime (i.e., at fixed box volume \(|\Lambda |\) and for a number of particles, N, sufficiently large), this method provides the same results in any dimension \(d\ge 1\). Furthermore, in the mean field limit, we can replace the ground state energy with the Bogoliubov energy in the expansion of the ground state vector.
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Notes
- 1.
We use the word mode both for \(\mathbf {j}\) and \(\frac{2\pi }{L}\mathbf {j}\)
- 2.
Concerning the validity in the thermodynamic limit of the replacement of \(a^*_{\mathbf {0}}a_{\mathbf {0}}\) with its average we refer the reader to Appendix D of [28].
- 3.
The operator \(\sum _{\mathbf {j}\in \mathbb {Z}^d\setminus \{\mathbf {0}\}}a^*_{\mathbf {j}}a_{\mathbf {j}}\) counts the number of particles in the nonzero modes states.
- 4.
We use this notation though the number of steps is in fact \(1+i/2\) being i an even number.
- 5.
\(\mathscr {K}^{Bog\,(i)}_{\mathbf {j}_*}(z)\) is self-adjoint on the domain of the Hamiltonian \(Q^{(>i+1)}_{\mathbf {j}_*}(H^{Bog}_{\mathbf {j}_*}-z)Q^{(>i+1)}_{\mathbf {j}_*}\).
- 6.
Due to point (b) in Theorem 4.6 of [32] reported above, for \(d\ge 3\) the (sufficiently large) density \(\rho \) can be chosen independently of L > 1 to ensure the existence of \(z_*\).
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Pizzo, A. (2018). Multi-scale Analysis in the Occupation Numbers of Particle States: An Application to Three-Modes Bogoliubov Hamiltonians. In: Cadamuro, D., Duell, M., Dybalski, W., Simonella, S. (eds) Macroscopic Limits of Quantum Systems. MaLiQS 2017. Springer Proceedings in Mathematics & Statistics, vol 270. Springer, Cham. https://doi.org/10.1007/978-3-030-01602-9_6
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