We 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.

%B Complex Analysis and Operator Theory %8 May %G eng %U https://doi.org/10.1007/s11785-019-00927-w %R 10.1007/s11785-019-00927-w %0 Report %D 2019 %T Zero modes and low-energy resolvent expansion for three dimensional Schrodinger operators with point interactions %A Raffaele Scandone %G eng %U https://arxiv.org/abs/1901.02449 %0 Journal Article %J Journal of Mathematical Physics %D 2018 %T Fractional powers and singular perturbations of quantum differential Hamiltonians %A Alessandro Michelangeli %A Andrea Ottolini %A Raffaele Scandone %XWe 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.

%B Journal of Mathematical Physics %V 59 %P 072106 %G eng %U https://doi.org/10.1063/1.5033856 %R 10.1063/1.5033856 %0 Journal Article %J Journal of Functional Analysis %D 2018 %T On fractional powers of singular perturbations of the Laplacian %A Vladimir Georgiev %A Alessandro Michelangeli %A Raffaele Scandone %K Point interactions %K Regular and singular component of a point-interaction operator %K Singular perturbations of the Laplacian %XWe 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.

%B Journal of Functional Analysis %V 275 %P 1551 - 1602 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022123618301046 %R https://doi.org/10.1016/j.jfa.2018.03.007 %0 Journal Article %J Zeitschrift für angewandte Mathematik und Physik %D 2018 %T Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials %A Paolo Antonelli %A Alessandro Michelangeli %A Raffaele Scandone %XWe 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.

%B Zeitschrift für angewandte Mathematik und Physik %V 69 %P 46 %8 Mar %G eng %U https://doi.org/10.1007/s00033-018-0938-5 %R 10.1007/s00033-018-0938-5 %0 Journal Article %J Annales Henri Poincaré %D 2018 %T Lp-Boundedness of Wave Operators for the Three-Dimensional Multi-Centre Point Interaction %A Gianfausto Dell'Antonio %A Alessandro Michelangeli %A Raffaele Scandone %A Kenji Yajima %XWe 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.

%B Annales Henri Poincaré %V 19 %P 283–322 %8 Jan %G eng %U https://doi.org/10.1007/s00023-017-0628-4 %R 10.1007/s00023-017-0628-4 %0 Report %D 2018 %T On real resonances for the three-dimensional, multi-centre point interaction %A Alessandro Michelangeli %A Raffaele Scandone %G eng %0 Journal Article %J Journal of Nonlinear Mathematical Physics %D 2018 %T Singular Hartree equation in fractional perturbed Sobolev spaces %A Alessandro Michelangeli %A Alessandro Olgiati %A Raffaele Scandone %XWe 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 Book Section %B Advances in Quantum Mechanics: Contemporary Trends and Open Problems %D 2017 %T Dispersive Estimates for Schrödinger Operators with Point Interactions in ℝ3 %A Felice Iandoli %A Raffaele Scandone %E Alessandro Michelangeli %E Gianfausto Dell'Antonio %XThe 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 $\mathbb{R}^3$, the perturbed Laplacian satisfies the same $L^p$−$L^q$ estimates of the free Laplacian in the smaller regime $q \in [2,3)$. These estimates are implied by a recent result concerning the Lpboundedness 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 $q \geq 3$.

%B Advances in Quantum Mechanics: Contemporary Trends and Open Problems %I Springer International Publishing %C Cham %P 187–199 %@ 978-3-319-58904-6 %G eng %U https://doi.org/10.1007/978-3-319-58904-6_11 %R 10.1007/978-3-319-58904-6_11