Bounds on the Bogoliubov--Hartree--Fock Energy of the Pauli--Fierz Hamiltonian

Abstract

A variational analysis of the Bogoliubov--Hartree--Fock (BHF) energy of the translation-invariant, spinless Pauli--Fierz Hamiltonian with massless dispersion relation built up on work of the first author, Breteaux, and Tzaneteas (2013) and of the first author and Hach (2022) is presented. The main results are lower and upper bounds on the BHF energy for fixed total momentum expressed through simpler variational problems defined on the space of positive Hilbert--Schmidt operators and a new variational formulation of the upper bound for zero total momentum. Specifically, we introduce a change of variables which considerably simplifies the energy functional and the derivation of its stationarity condition.