@article {Michelangeli2019, title = {Point-Like Perturbed Fractional Laplacians Through Shrinking Potentials of Finite Range}, journal = {Complex Analysis and Operator Theory}, year = {2019}, month = {May}, abstract = {

We construct the rank-one, singular (point-like) perturbations of the d-dimensional fractional Laplacian in the physically meaningful norm-resolvent limit of fractional Schr{\"o}dinger 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 Schr{\"o}dinger operators formed by a fractional Laplacian and a regular potential.

}, issn = {1661-8262}, doi = {10.1007/s11785-019-00927-w}, url = {https://doi.org/10.1007/s11785-019-00927-w}, author = {Alessandro Michelangeli and Raffaele Scandone} } @article {1901.02449, title = {Zero modes and low-energy resolvent expansion for three dimensional Schrodinger operators with point interactions}, year = {2019}, url = {https://arxiv.org/abs/1901.02449}, author = {Raffaele Scandone} } @article {doi:10.1063/1.5033856, title = {Fractional powers and singular perturbations of quantum differential Hamiltonians}, journal = {Journal of Mathematical Physics}, volume = {59}, number = {7}, year = {2018}, pages = {072106}, abstract = {

We 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 Schr{\"o}dinger equations for the corresponding operators, we outline a programme of relevant questions that deserve being investigated.

}, doi = {10.1063/1.5033856}, url = {https://doi.org/10.1063/1.5033856}, author = {Alessandro Michelangeli and Andrea Ottolini and Raffaele Scandone} } @article {GEORGIEV20181551, title = {On fractional powers of singular perturbations of the Laplacian}, journal = {Journal of Functional Analysis}, volume = {275}, number = {6}, year = {2018}, pages = {1551 - 1602}, abstract = {

We 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.

}, keywords = {Point interactions, Regular and singular component of a point-interaction operator, Singular perturbations of the Laplacian}, issn = {0022-1236}, doi = {https://doi.org/10.1016/j.jfa.2018.03.007}, url = {http://www.sciencedirect.com/science/article/pii/S0022123618301046}, author = {Vladimir Georgiev and Alessandro Michelangeli and Raffaele Scandone} } @article {Antonelli2018, title = {Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials}, journal = {Zeitschrift f{\"u}r angewandte Mathematik und Physik}, volume = {69}, number = {2}, year = {2018}, month = {Mar}, pages = {46}, abstract = {

We prove the existence of weak solutions in the space of energy for a class of nonlinear Schr{\"o}dinger 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.

}, issn = {1420-9039}, doi = {10.1007/s00033-018-0938-5}, url = {https://doi.org/10.1007/s00033-018-0938-5}, author = {Paolo Antonelli and Alessandro Michelangeli and Raffaele Scandone} } @article {Dell{\textquoteright}Antonio2018, title = {Lp-Boundedness of Wave Operators for the Three-Dimensional Multi-Centre Point Interaction}, journal = {Annales Henri Poincar{\'e}}, volume = {19}, number = {1}, year = {2018}, month = {Jan}, pages = {283{\textendash}322}, abstract = {

We prove that, for arbitrary centres and strengths, the wave operators for three-dimensional Schr{\"o}dinger operators with multi-centre local point interactions are bounded in Lp(R3)for 1\<p\<3 and unbounded otherwise.

}, issn = {1424-0661}, doi = {10.1007/s00023-017-0628-4}, url = {https://doi.org/10.1007/s00023-017-0628-4}, author = {Gianfausto Dell{\textquoteright}Antonio and Alessandro Michelangeli and Raffaele Scandone and Kenji Yajima} } @article {2018, title = {On real resonances for the three-dimensional, multi-centre point interaction}, number = {SISSA;40/2018/MATE}, year = {2018}, author = {Alessandro Michelangeli and Raffaele Scandone} } @article {doi:10.1080/14029251.2018.1503423, title = {Singular Hartree equation in fractional perturbed Sobolev spaces}, journal = {Journal of Nonlinear Mathematical Physics}, volume = {25}, number = {4}, year = {2018}, pages = {558-588}, publisher = {Taylor \& Francis}, abstract = {

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 Schr{\"o}dinger 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.

}, doi = {10.1080/14029251.2018.1503423}, url = {https://doi.org/10.1080/14029251.2018.1503423}, author = {Alessandro Michelangeli and Alessandro Olgiati and Raffaele Scandone} } @inbook {Iandoli2017, title = {Dispersive Estimates for Schr{\"o}dinger Operators with Point Interactions in ℝ3}, booktitle = {Advances in Quantum Mechanics: Contemporary Trends and Open Problems}, year = {2017}, pages = {187{\textendash}199}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, abstract = {

The study of dispersive properties of Schr{\"o}dinger 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 Schr{\"o}dinger 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 $q \geq 3$.

}, isbn = {978-3-319-58904-6}, doi = {10.1007/978-3-319-58904-6_11}, url = {https://doi.org/10.1007/978-3-319-58904-6_11}, author = {Felice Iandoli and Raffaele Scandone}, editor = {Alessandro Michelangeli and Gianfausto Dell{\textquoteright}Antonio} }