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

Alessandro Pizzo
(University of Rome, Tor Vergata)
Mat/Fys-seminar
Torsdag, 25 oktober, 2018, at 16:15-17:00, in Aud. D3 (1531-215)
Abstrakt:
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.
Kontaktperson: Jacob Schach Møller