Point-like perturbed fractional Laplacians through shrinking potentials of finite range

We reconstruct 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 the 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

Fractional powers and singular perturbations of quantum differential Hamiltonians

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 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\"{o}dinger equations for the corresponding operators we outline a programme of relevant questions that deserve being investigated.Comment: Published on J. Math. Phys. (2018

The singular Hartree equation in fractional perturbed Sobolev spaces

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\"odinger 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.Comment: Published on Journal of Nonlinear Mathematical Physics (2018

On fractional powers of singular perturbations of the Laplacian

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

Non-linear Schrodinger equations with singular perturbations and with rough magnetic potentials

In this thesis we discuss thoroughly a class of linear and non-linear Schrodinger equations that arise in various physical contexts of modern relevance. First we work in the scenario where the main linear part of the equation is a singular perturbation of a symmetric pseudo-differential operator, which formally amounts to add to it a potential supported on a finite set of points. A detailed discussion on the rigorous realisations and the main properties of such objects is given when the unperturbed pesudo-differential operator is the fractional Laplacian on R^d. We then consider the relevant special case of singular perturbations of the three-dimensional non-fractional Laplacian: we qualify their smoothing and scattering properties, and characterise their fractional powers and induced Sobolev norms. As a consequence, we are able to establish local and global solution theory for a class of singular Schrodinger equations with convolution-type non-linearity. As a second main playground, we consider non-linear Schrodinger equations with time-dependent, rough magnetic fields, and with local and non-local non-linearities. We include magnetic fields for which the corresponding Strichartz estimates are not available. To this aim, we introduce a suitable parabolic regularisation in the magnetic Laplacian: by exploiting the smoothing properties of the heat-Schrodinger propagator and the mass/energy bounds, we are able to construct global solutions for the approximated problem. Finally, through a compactness argument, we can remove the regularisation and deduce the existence of global, finite energy, weak solutions to the original equation

Estudio de la pubertad adelantada y sus variables en atención primaria

La Pubertad Adelantada es una variante de la normalidad de la Pubertad que se define como la aparición de caracteres sexuales secundarios entre los 8 y los 9 años en las niñas y entre los 9 y los 10 en los niños. Es una condición benigna y el tratamiento se decide en caso de que se evidencie patología subyacente, rápido progreso en la aparición de los caracteres sexuales secundarios o indicación psicológica. Se ha relacionado con factores nutricionales así como somatometría perinatal.Grado en Medicin