Low energy behavior in few-particle quantum systems: Efimov effect and zero-range interactions

We investigate the emergence of a universal behavior in certain few-particle quantum sys- tems at low-energy. First we consider a system composed by two identical fermions of mass one and a dif- ferent particle of mass m in dimension three. Under the assumption that the two-particle Hamiltonians composed by one of the fermions and the third particle have a resonance at zero-energy, and for m less than a mass threshold m∗, we prove the occurrence of the Efimov effect, i. e., the existence of an infinite number of three-body bound states accumulating at zero. Then we study three-particle systems with zero-range interactions. In dimension one we give a rigorous definition of the Hamiltonian for three identical bosons and we prove that it is the limit of suitably rescaled regular Hamiltonians. In dimension three we write the expression of the quadratic form associated to the STM extension for a generic three-particle system. Then we focus on a system of three identical bosons proving stability outside the s-wave subspace. As a third example of universal behavior in few-particle system we con- sider a quantum Lorentz gas in dimension three: a particle moving through N obstacles whose positions are independently chosen according to a given common probability density. We assume that the particle interact with each obstacle via a Gross Pitaevskii potential. We prove the convergence, as N → ∞, to a Hamiltonian depending on the common distribution density of the obstacles and such that the only dependence on the interaction potential is through its scattering length.