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.

Non-linear Schrodinger equations with singular perturbations and with rough magnetic potentials / Scandone, Raffaele. - (2018 Sep 26).

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

Scandone, Raffaele
2018

Abstract

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.
Michelangeli, Alessandro
Scandone, Raffaele
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/82770
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