00956nas a2200133 4500008004100000245008600041210006900127260001000196520049500206100002900701700002100730700002300751856004800774 2018 en d00aFractional powers and singular perturbations of quantum differential Hamiltonians0 aFractional powers and singular perturbations of quantum differen bSISSA3 aWe 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 such two 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.1 aMichelangeli, Alessandro1 aOttolini, Andrea1 aScandone, Raffaele uhttp://preprints.sissa.it/handle/1963/3530501050nas a2200121 4500008004100000245009200041210006900133260001000202520061600212100002900828700002300857856004800880 2018 en d00aPoint-like perturbed fractional Laplacians through shrinking potentials of finite range0 aPointlike perturbed fractional Laplacians through shrinking pote bSISSA3 aWe 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.1 aMichelangeli, Alessandro1 aScandone, Raffaele uhttp://preprints.sissa.it/handle/1963/3531301196nas a2200109 4500008004100000245008200041210007000123520080200193100002000995700002301015856004801038 2017 en d00aDispersive estimates for Schrödinger operators with point interactions in R30 aDispersive estimates for Schrödinger operators with point intera3 aThe 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 ≥ 3.1 aIandoli, Felice1 aScandone, Raffaele uhttp://preprints.sissa.it/handle/1963/3527700952nas a2200121 4500008004100000245006800041210006500109520053300174100002300707700002900730700002300759856004800782 2017 en d00aOn fractional powers of singular perturbations of the Laplacian0 afractional powers of singular perturbations of the Laplacian3 aWe 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.1 aGeorgiev, Vladimir1 aMichelangeli, Alessandro1 aScandone, Raffaele uhttp://preprints.sissa.it/handle/1963/3529301194nas a2200121 4500008004100000245010100041210006900142520074000211100002100951700002900972700002301001856004801024 2017 en d00aGlobal, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials0 aGlobal finite energy weak solutions for the NLS with rough timed3 aWe 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.1 aAntonelli, Paolo1 aMichelangeli, Alessandro1 aScandone, Raffaele uhttp://preprints.sissa.it/handle/1963/3529400711nas a2200133 4500008004100000245009400041210006900135520022600204100002900430700002900459700002300488700001800511856004800529 2017 en d00aLp-boundedness of wave operators for the three-dimensional multi-centre point interaction0 aLpboundedness of wave operators for the threedimensional multice3 aWe 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.1 aDell'Antonio, Gianfausto1 aMichelangeli, Alessandro1 aScandone, Raffaele1 aYajima, Kenji uhttp://preprints.sissa.it/handle/1963/3528301216nas a2200121 4500008004100000245006900041210006900110520079100179100002900970700002400999700002301023856004801046 2017 en d00aSingular Hartree equation in fractional perturbed Sobolev spaces0 aSingular Hartree equation in fractional perturbed Sobolev spaces3 aWe 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.1 aMichelangeli, Alessandro1 aOlgiati, Alessandro1 aScandone, Raffaele uhttp://preprints.sissa.it/handle/1963/35301