TY - RPRT
T1 - Fractional powers and singular perturbations of differential operators
Y1 - 2018
A1 - Alessandro Michelangeli
A1 - Andrea Ottolini
A1 - Raffaele Scandone
AB - We 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.
PB - SISSA
UR - http://preprints.sissa.it/handle/1963/35305
N1 - Dedicated to Gianfausto Dell'Antonio on the occasion of his 85th birthday
U1 - 35611
U2 - Mathematics
U4 - 1
ER -
TY - RPRT
T1 - Dispersive estimates for Schrödinger operators with point interactions in R3
Y1 - 2017
A1 - Felice Iandoli
A1 - Raffaele Scandone
AB - The 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.
UR - http://preprints.sissa.it/handle/1963/35277
U1 - 35584
U2 - Mathematics
U4 - 1
ER -
TY - RPRT
T1 - On fractional powers of singular perturbations of the Laplacian
Y1 - 2017
A1 - Vladimir Georgiev
A1 - Alessandro Michelangeli
A1 - Raffaele Scandone
AB - 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, 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.
UR - http://preprints.sissa.it/handle/1963/35293
N1 - Partially supported by the 2014-2017 MIUR-FIR grant \Cond-Math: Condensed Matter and
Mathematical Physics" code RBFR13WAET.
U1 - 35599
U2 - Mathematics
ER -
TY - RPRT
T1 - Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials
Y1 - 2017
A1 - Paolo Antonelli
A1 - Alessandro Michelangeli
A1 - Raffaele Scandone
AB - We 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.
UR - http://preprints.sissa.it/handle/1963/35294
U1 - 35600
U2 - Mathematics
ER -
TY - RPRT
T1 - The Lp-boundedness of wave operators for the three-dimensional multi-centre point interaction
Y1 - 2017
A1 - Gianfausto Dell'Antonio
A1 - Alessandro Michelangeli
A1 - Raffaele Scandone
A1 - Kenji Yajima
AB - We 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.
UR - http://preprints.sissa.it/handle/1963/35283
U1 - 35588
U2 - Mathematics
U4 - 1
ER -
TY - RPRT
T1 - The Singular Hartree Equation in Fractional Perturbed Sobolev Spaces
Y1 - 2017
A1 - Alessandro Michelangeli
A1 - Alessandro Olgiati
A1 - Raffaele Scandone
AB - 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 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.
UR - http://preprints.sissa.it/handle/1963/35301
U1 - 35607
U2 - Mathematics
U4 - 1
ER -