# Multi-scale analysis in the occupation numbers of particle states: an application to three-modes Bogoliubov Hamiltonians

Alessandro Pizzo (University of Rome, Tor Vergata)
Torsdag 25. oktober 2018 16:15–17:00 Aud. D3 (1531-215)
Mat/Fys-seminar
I shall report on a multi-scale technique introduced 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 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 $d$ larger than or equal to $3$, for an arbitrarily large box and at fixed, large particle density, 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.

If time permits I will also explain similar results for the complete Hamiltonian describing two-body interactions between the gas particles in the mean field limiting regime.
Kontakt: Jacob Schach Møller Revideret: 18.10.2018