A rigorous derivation of the Bogoliubov-Gross-Pitaevskii equation for superfluids

In the present Thesis, a rigorous derivation of the Bogoliubov-Gross-Pitaevskii (BGP) equation, which completely describes the dynamics of the condensed phase of a boson fluid at zero temperature, is presented. The problem is dealt within the formalism of second quantization, with an external trapping potential working as a vessel, the well amplitude size ruling the large size limit of the system (at a fixed density). To begin with, in the First Chapter, the minimal necessary scaling hypotheses are discussed and compared with both the theoretical and the experimental ones existing in the literature. This is relevant in a problem where the existence of an effective equation in the thermodynamic limit almost always requires to let some physical parameters characterizing the system (e.g. the range of the two body potential) to depend on the size of the latter. Once determined the right scaling regime, one is left with a problem in dimensionless form where, essentially, the dynamics of the boson quantum field is proven to be close to that of a problem where the two-body potential is delta-like, multiplied by a coupling constant that is explicitly computed in terms of all the parametres of the system (such as number density, two-body interaction, and so on). On the other hand, the fundamental boson commutation rules satisfied by the rescaled quantum field are of the semi-classical form, with a commutator that vanishes in the large size limit. At this stage, by analogy with what has been done for the finite version of the problem, i.e. for Bose-Hubbard models, in the Second Chapter we take the expectation of the quantum field on a coherent state distributed according to a quantum invariant Gaussian thermal measure. Such an expectation, or Wick symbol, defines the scalar field that satisfies the BGP equation in a suitable infrared limit. Finally, the convergence of the time-dependent Wick symbol defined above to the solution of the scalar BGP equation, in measure norm, is proven; specifically, the bounding constant of the distance is found to be depending linearly on time. Further, it is shown that such constant goes to zero in the thermodynamic limit, thus ensuring the exact convergence to BGP scalar dynamics at zero temperature, though a condition needs to be satisfied by the trapping potential.