SDLThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.http://preprints.sissa.it:80/xmlui2017-11-16T06:28:16Z2017-11-16T06:28:16ZThe Singular Hartree Equation in Fractional Perturbed Sobolev SpacesMichelangeli, AlessandroOlgiati, AlessandroScandone, Raffaelehttp://preprints.sissa.it/handle/1963/353012017-11-15T01:00:32Z2017-01-01T00:00:00ZThe Singular Hartree Equation in Fractional Perturbed Sobolev Spaces
Michelangeli, Alessandro; Olgiati, Alessandro; Scandone, Raffaele
We establish the local and global theory for the Cauchy problem
of the singular Hartree equation in three dimensions, that is, the modification
of the non-linear SchrÃ¶dinger equation with Hartree non-linearity, where the
linear part is now given by the Hamiltonian of point interaction. The latter is
a singular, self-adjoint perturbation of the free Laplacian, modelling a contact
interaction at a fixed point. The resulting non-linear equation is the typical
effective equation for the dynamics of condensed Bose gases with fixed pointlike
impurities. We control the local solution theory in the perturbed Sobolev
spaces of fractional order between the mass space and the operator domain.
We then control the global solution theory both in the mass and in the energy
space.
2017-01-01T00:00:00ZDiscrete spectra for critical Dirac-Coulomb HamiltoniansGallone, MatteoMichelangeli, Alessandrohttp://preprints.sissa.it/handle/1963/353002017-11-07T01:00:33Z2017-01-01T00:00:00ZDiscrete spectra for critical Dirac-Coulomb Hamiltonians
Gallone, Matteo; Michelangeli, Alessandro
The one-particle Dirac Hamiltonian with Coulomb interaction is known to be realised,
in a regime of large (critical) couplings, by an infinite multiplicity of distinct
self-adjoint operators, including a distinguished, physically most natural one. For
the latter, Sommerfeld's celebrated fine structure formula provides the well-known
expression for the eigenvalues in the gap of the continuum spectrum. Exploiting
our recent general classification of all other self-adjoint realisations, we generalise
Sommerfeld's formula so as to determine the discrete spectrum of all other selfadjoint
versions of the Dirac-Coulomb Hamiltonian. Such discrete spectra display
naturally a fibred structure, whose bundle covers the whole gap of the continuum
spectrum.
2017-01-01T00:00:00ZA uniqueness result for the decomposition of vector fields in RdBianchini, StefanoBonicatto, Paolohttp://preprints.sissa.it/handle/1963/352992017-10-24T00:00:35Z2017-01-01T00:00:00ZA uniqueness result for the decomposition of vector fields in Rd
Bianchini, Stefano; Bonicatto, Paolo
2017-01-01T00:00:00ZOn contact interactions realised as Friedrichs systemsErceg, MarcoMichelangeli, Alessandrohttp://preprints.sissa.it/handle/1963/352982017-10-11T00:00:32Z2017-01-01T00:00:00ZOn contact interactions realised as Friedrichs systems
Erceg, Marco; Michelangeli, Alessandro
We study the Hamiltonians of contact (or `point', or `zero-range') interactions in
quantum mechanics via the framework of abstract Friedrichs systems. More precisely, we study one- and three-dimensional case of contact interactions, and we present a new method how to obtain all self-adjoint (even closed) realisations of interest. In particular, in the one-dimensional case we recover both gamma and gamma' interactions.
2017-01-01T00:00:00Z