Browsing by Author "Scandone, Raffaele"
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Item Dispersive estimates for Schrödinger operators with point interactions in R3(2017-03-30) Iandoli, Felice; Scandone, Raffaele; MathematicsThe study of dispersive properties of Schrödinger operators with point interactions is a fundamental tool for understanding the behavior of many body quantum systems interacting with very short range potential, whose dynamics can be approximated by non linear Schrödinger equations with singular interactions. In this work we proved that, in the case of one point interaction in R3, the perturbed Laplacian satisfies the same Lp -Lq estimates of the free Laplacian in the smaller regime q ∈ 2 [2;3). These estimates are implied by a recent result concerning the Lp boundedness of the wave operators for the perturbed Laplacian. Our approach, however, is more direct and relatively simple, and could potentially be useful to prove optimal weighted estimates also in the regime q ≥ 3.Item Fractional powers and singular perturbations of quantum differential Hamiltonians(SISSA, 2018-01) Michelangeli, Alessandro; Ottolini, Andrea; Scandone, Raffaele; MathematicsWe consider the fractional powers of singular (point-like) perturbations of the Laplacian and the singular perturbations of fractional powers of the Laplacian, and we compare two such constructions focusing on their perturbative structure for resolvents and on the local singularity structure of their domains. In application to the linear and non-linear Schrödinger equations for the corresponding operators, we outline a programme of relevant questions that deserve being investigated.Item Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials(2017) Antonelli, Paolo; Michelangeli, Alessandro; Scandone, Raffaele; MathematicsWe prove the existence of weak solutions in the space of energy for a class of nonlinear Schrödinger equations in the presence of a external, rough, time-dependent magnetic potential. Under our assumptions, it is not possible to study the problem by means of usual arguments like resolvent techniques or Fourier integral operators, for example. We use a parabolic regularisation, and we solve the approximating Cauchy problem. This is achieved by obtaining suitable smoothing estimates for the dissipative evolution. The total mass and energy bounds allow to extend the solution globally in time. We then infer sufficient compactness properties in order to produce a global-in-time finite energy weak solution to our original problem.Item Lp-boundedness of wave operators for the three-dimensional multi-centre point interaction(2017-04-13) Dell'Antonio, Gianfausto; Michelangeli, Alessandro; Scandone, Raffaele; Yajima, Kenji; MathematicsWe prove that, for arbitrary centres and strengths, the wave operators for three dimensional Schrödinger operators with multi-centre local point interactions are bounded in Lp(R3) for 1 < p < 3 and unbounded otherwise.Item On fractional powers of singular perturbations of the Laplacian(2017) Georgiev, Vladimir; Michelangeli, Alessandro; Scandone, Raffaele; MathematicsWe qualify a relevant range of fractional powers of the so-called Hamiltonian of point interaction in three dimensions, namely the singular perturbation of the negative Laplacian with a contact interaction supported at the origin. In particular we provide an explicit control of the domain of such a fractional operator and of its decomposition into regular and singular parts. We also qualify the norms of the resulting singular fractional Sobolev spaces and their mutual control with the corresponding classical Sobolev norms.Item On real resonances for three-dimensional Schrödinger operators with point interactions(SISSA, 2019-02-20) Michelangeli, Alessandro; Scandone, RaffaeleWe prove the absence of positive real resonances for Schroedinger operators with finitely many point interactions in R3 and we discuss such a property from the perspective of dispersive and scattering features of the associated Schr¨odinger propagator.Item On real resonances for three-dimensional Schrodinger operators with point ¨ interactions†(2018) Michelangeli, Alessandro; Scandone, Raffaele; mathematicsWe prove the absence of positive real resonances for Schrodinger operators with finitely ¨ many point interactions in R 3 and we discuss such a property from the perspective of dispersive and scattering features of the associated Schrodinger propagator.Item Point-like perturbed fractional Laplacians through shrinking potentials of finite range(SISSA, 2018-03) Michelangeli, Alessandro; Scandone, Raffaele; MathematicsWe construct the rank-one, singular (point-like) perturbations of the d-dimensional fractional Laplacian in the physically meaningful norm-resolvent limit of fractional Schrödinger operators with regular potentials centred around the perturbation point and shrinking to a delta-like shape. We analyse both possible regimes, the resonance-driven and the resonance-independent limit, depending on the power of the fractional Laplacian and the spatial dimension. To this aim, we also qualify the notion of zero-energy resonance for Schrödinger operators formed by a fractional Laplacian and a regular potential.Item Singular Hartree equation in fractional perturbed Sobolev spaces(2017) Michelangeli, Alessandro; Olgiati, Alessandro; Scandone, Raffaele; MathematicsWe 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 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.Item Zero modes and low-energy resolvent expansion for three dimensional Schrodinger operators with point interactions(SISSA, 2019) Scandone, RaffaeleWe study the low energy behavior of the resolvent of Schrodinger operators with finitely many point interactions in three dimensions. We also discuss the occurrence and the multiplicity of zero energy obstructions.