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 Schrödinger 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 point-like 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.

%B Journal of Nonlinear Mathematical Physics %I Taylor & Francis %V 25 %P 558-588 %G eng %U https://doi.org/10.1080/14029251.2018.1503423 %R 10.1080/14029251.2018.1503423 %0 Report %D 2017 %T Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei %A Matteo Gallone %A Alessandro Michelangeli %X We derive a classification of the self-adjoint extensions of the three-dimensional Dirac-Coulomb operator in the critical regime of the Coulomb coupling. Our approach is solely based upon the KreĬn-Višik- Birman extension scheme, or also on Grubb's universal classification theory, as opposite to previous works within the standard von Neu- mann framework. This let the boundary condition of self-adjointness emerge, neatly and intrinsically, as a multiplicative constraint between regular and singular part of the functions in the domain of the exten- sion, the multiplicative constant giving also immediate information on the invertibility property and on the resolvent and spectral gap of the extension. %G en %U http://preprints.sissa.it/handle/1963/35287 %1 35592 %2 Mathematics %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-06-05T06:42:39Z No. of bitstreams: 1 Coulomb_Gallone_Michelangeli_26_2017.pdf: 508858 bytes, checksum: 21c788d0de4fed88bee2e6fd5cd0849c (MD5) %0 Journal Article %D 2017 %T Spectral Properties of the 2+1 Fermionic Trimer with Contact Interactions %A Simon Becker %A Alessandro Michelangeli %A Andrea Ottolini %X We qualify the main features of the spectrum of the Hamiltonian of point interaction for a three-dimensional quantum system consisting of three point-like particles, two identical fermions, plus a third particle of different species, with two-body interaction of zero range. For arbitrary magnitude of the interaction, and arbitrary value of the mass parameter (the ratio between the mass of the third particle and that of each fermion) above the stability threshold, we identify the essential spectrum, localise and prove the finiteness of the discrete spectrum, qualify the angular symmetry of the eigenfunctions, and prove the monotonicity of the eigenvalues with respect to the mass parameter. We also demonstrate the existence of bound states in a physically relevant regime of masses. %I SISSA %G en %U http://preprints.sissa.it/handle/1963/35303 %1 35609 %2 Mathematics %4 1 %$ Submitted by mmarin@sissa.it (mmarin@sissa.it) on 2018-01-04T08:57:37Z No. of bitstreams: 1 SISSA_preprint_61-2017-MATE.pdf: 465833 bytes, checksum: bc2bbd7578ccfcd69ca2a33a2fc3adb0 (MD5) %0 Report %D 2015 %T Schödinger operators on half-line with shrinking potentials at the origin %A Gianfausto Dell'Antonio %A Alessandro Michelangeli %X We discuss the general model of a Schrödinger quantum particle constrained on a straight half-line with given self-adjoint boundary condition at the origin and an interaction potential supported around the origin. We study the limit when the range of the potential scales to zero and its magnitude blows up. We show that in the limit the dynamics is generated by a self-adjoint negative Laplacian on the half-line, with a possible preservation or modification of the boundary condition at the origin, depending on the magnitude of the scaling and of the strength of the potential. %I SISSA %G en %U http://urania.sissa.it/xmlui/handle/1963/34439 %1 34566 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2015-02-09T07:28:40Z No. of bitstreams: 1 SISSA_preprint_06-2015-MATE.pdf: 177243 bytes, checksum: 383296953d36b4cec0cb4cfb55e95d7f (MD5) %0 Report %D 2015 %T Stability of closed gaps for the alternating Kronig-Penney Hamiltonian %A Alessandro Michelangeli %A Domenico Monaco %X We consider the Kronig-Penney model for a quantum crystal with equispaced periodic delta-interactions of alternating strength. For this model all spectral gaps at the centre of the Brillouin zone are known to vanish, although so far this noticeable property has only been proved through a very delicate analysis of the discriminant of the corresponding ODE and the associated monodromy matrix. We provide a new, alternative proof by showing that this model can be approximated, in the norm resolvent sense, by a model of regular periodic interactions with finite range for which all gaps at the centre of the Brillouin zone are still vanishing. In particular this shows that the vanishing gap property is stable in the sense that it is present also for the "physical" approximants and is not only a feature of the idealised model of zero-range interactions. %I SISSA %G en %U http://urania.sissa.it/xmlui/handle/1963/34460 %1 34629 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2015-04-01T14:57:03Z No. of bitstreams: 1 SISSA_preprint_16-2015-MATE.pdf: 349807 bytes, checksum: f5e6dacd85fb41c96420795cc37cc12c (MD5) %0 Report %D 2015 %T Stability of the (2+2)-fermionic system with zero-range interaction %A Alessandro Michelangeli %A Paul Pfeiffer %X We introduce a 3D model, and we study its stability, consisting of two distinct pairs of identical fermions coupled with a two-body interaction between fermions of different species, whose effective range is essentially zero (a so called (2+2)-fermionic system with zero-range interaction). The interaction is modelled by implementing the the celebrated (and ubiquitous, in the literature of this field) Bethe-Peierls contact condition with given two-body scattering length within the Krein-Visik-Birman theory of extensions of semi-bounded symmetric operators, in order to make the Hamiltonian a well-defined (self-adjoint) physical observable. After deriving the expression for the associated energy quadratic form, we show analytically and numerically that the energy of the model is bounded below, thus describing a stable system. %G en %U http://urania.sissa.it/xmlui/handle/1963/34474 %1 34649 %2 Mathematics %4 1 %# MAT/07 %$ Submitted by Alessandro Michelangeli (alemiche@sissa.it) on 2015-06-24T10:47:38Z No. of bitstreams: 1 sissa-preprint-29-2015-mate.pdf: 827153 bytes, checksum: e5268c2d26f348929a3c532f9ffd3097 (MD5) %0 Journal Article %J Rev. Math. Phys. 24 (2012), 1250017 %D 2012 %T Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions %A Michele Correggi %A Gianfausto Dell'Antonio %A Domenico Finco %A Alessandro Michelangeli %A Alessandro Teta %X We study the stability problem for a non-relativistic quantum system in\\r\\ndimension three composed by $ N \\\\geq 2 $ identical fermions, with unit mass,\\r\\ninteracting with a different particle, with mass $ m $, via a zero-range\\r\\ninteraction of strength $ \\\\alpha \\\\in \\\\R $. We construct the corresponding\\r\\nrenormalised quadratic (or energy) form $ \\\\form $ and the so-called\\r\\nSkornyakov-Ter-Martirosyan symmetric extension $ H_{\\\\alpha} $, which is the\\r\\nnatural candidate as Hamiltonian of the system. We find a value of the mass $\\r\\nm^*(N) $ such that for $ m > m^*(N)$ the form $ \\\\form $ is closed and bounded from below. As a consequence, $ \\\\form $ defines a unique self-adjoint and bounded from below extension of $ H_{\\\\alpha}$ and therefore the system is stable. On the other hand, we also show that the form $ \\\\form $ is unbounded from below for $ m < m^*(2)$. In analogy with the well-known bosonic case, this suggests that the system is unstable for $ m < m^*(2)$ and the so-called Thomas effect occurs. %B Rev. Math. Phys. 24 (2012), 1250017 %I World Scientific %G en %U http://hdl.handle.net/1963/6069 %1 5955 %2 Mathematics %3 Mathematical Physics %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2012-08-02T07:19:59Z\\nNo. of bitstreams: 1\\n1201.5740v1.pdf: 303047 bytes, checksum: eb2df0afd547514c235422e82f494584 (MD5) %R 10.1142/S0129055X12500171 %0 Report %D 2007 %T Strengthened convergence of marginals to the cubic nonlinear Schroedinger equation %A Alessandro Michelangeli %X We rewrite a recent derivation of the cubic non-linear Schroedinger equation by Adami, Golse, and Teta in the more natural formof the asymptotic factorisation of marginals at any fixed time and in the trace norm. This is the standard form in which the emergence of the\\nnon-linear effective dynamics of a large system of interacting bosons is\\nproved in the literature. %G en_US %U http://hdl.handle.net/1963/1977 %1 2218 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-06-06T09:21:56Z\\nNo. of bitstreams: 1\\nstrengthened.pdf: 228368 bytes, checksum: c87b1b3ae8fd722412cf906c5f44df5b (MD5)