Ricerc@Sapienza

We present two possible strategies to obtain a lower bounded Hamiltonian for three bosons interacting through zero-range interactions. First, we investigate a family of zero-range Hamiltonians defined in a Hilbert space of tensorial wave functions. Then, we examine the regularizing effect of an ultraviolet cutoff on the boundary conditions satisfied by functions in the form domain of zero-range Hamiltonians of a three boson system.

Weinvestigatethelocalizationpropertiesofindependentelectronsinaperi- odic background, possibly including a periodic magnetic field, as e. g. in Chern insulators and in quantum Hall systems. Since, generically, the spectrum of the Hamiltonian is absolutely continuous, localization is characterized by the decay, as |x| → ∞, of the composite (magnetic) Wannier functions associated to the Bloch bands below the Fermi energy, which is supposed to be in a spectral gap.