Convergence to Hartree dynamics for interacting bosons

The main goal of the present thesis is to derive a rigorous estimate for the convergence to Hartree dynamics for interacting bosons in the low effective temperature limit. The 2-body interaction potential is chosen in the Hardy class and we allow our potential to have negative values, in order to model also attractive forces between particles. The estimate is obtained working on proper functional norms on the space of square-integrable functions, endowed with a thermal gaussian measure μ that concentrates around the ground state in the low effective temperature limit. We write the normal mode decomposition of the quantum field operators, and then truncate it by introducing an UV cutoff Λ. The cutoff is introduced to switch from the infinitely-many coupled ODEs describing the evolution of the annihilation operator to a finite ODEs system. The dynamics of the system with the cutoff is studied on the Bargmann-Fock space, a subspace of the bosonic Fock space in the second quantization formalism. We use coherent state expectation values to obtain scalar equations starting from operatorial ones. More precisely, coherent states are introduced algebraically through the action of the Weyl-Heisenberg translation operator; then the Bargmann transform and the corresponding Bargmann representation are introduced. We use the Bargmann representation of the canonical coherent states to compute the Wick symbol of the operators (i.e. the coherent expectation value). Wick symbols are also used to define the μ-norm for operators. Finally, a bound on the distance (through the above mentioned norms) between the regularized and the full quantum dynamics is provided, and the bound dependence on cutoff, effective temperature and time is explicitly shown. Remarkably, we find linear time dependence for the bound.