00984nas a2200133 4500008004100000245007500041210006900116260001000185520053400195100002900729700002100758700002300779856004800802 2018 en d00aFractional powers and singular perturbations of differential operators0 aFractional powers and singular perturbations of differential ope bSISSA3 aWe develop the construction of the fractional powers of singular
(point-like) perturbations of the Laplacian, and the construction of 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 Schr odinger 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/1963/3530501196nas 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/1963/3527700972nas a2200121 4500008004100000245006800041210006500109520055300174100002300727700002900750700002300779856004800802 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, when applicable, of the 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/1963/3529301184nas a2200121 4500008004100000245010100041210006900142520073000211100002100941700002900962700002300991856004801014 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 non-linear Schördinger equations in the presence of a external rough
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 regularization 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 suffcient
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/1963/3529400715nas a2200133 4500008004100000245009800041210006900139520022600208100002900434700002900463700002300492700001800515856004800533 2017 en d00aThe Lp-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/1963/3528301220nas a2200121 4500008004100000245007300041210006900114520079100183100002900974700002401003700002301027856004801050 2017 en d00aThe Singular 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