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Eceizabarrena,
Daniel
Thesis
The Pointwise Convergence of the Solution to the Schrödinger Equation
Date:
27062016
University:
Universidad Autónoma de Madrid (UAM)
Country:
Spain
File:
Download PDF
Comments:
The present document corresponds to the subject
Trabajo Fin de Máster
being part of the
Master's Degree in Mathematics and Applications
at Universidad Autónoma de Madrid. The objective of this dissertation is to analyse the almost everywhere convergence of the solution to the Schrödinger equation to the given initial data.
This problem was presented by Lennart Carleson in 1980, when he proved convergence for the one space dimensional problem and for data lying in Sobolev spaces
$$
H
^{s}
with exponents
$s\; \ge \; 1/4$
. Since then, many partial results have shown that Sobolev spaces are the convenient setup to work with. The problem was solved in one dimension in 1981 by B.E.J. Dahlberg and C.E. Kenig, who showed Carleson's condition was also necessary. We review both results in the first chapter.
The question still remains unsolved for higher dimensions. Throughout the rest of the chapters, we present several partial results, particularly the best necessary and sufficient conditions known so far.
Distribution Theory and Fundamental Solutions of Differential Operators
Date:
24062015
University:
Euskal Herriko Unibertsitatea (UPV/EHU)
Country:
Spain
File:
Download PDF
Comments:
This work is the written document of the subject
Gradu Amaierako Lana
, the final thesis corresponding to the
Degree in Mathematics
.
The objective of this dissertation is to study the basics of the
theory of distributions
and show how it can be used to obtain what we call fundamental solutions of partial differential equations. The first two chapters are devoted to describe several spaces of test functions and distributions. The third chapter covers the Fourier transform, which gives a way to connect distributions and partial differential equations. In the last chapter we work to obtain some fundamental solutions, such as the ones for the heat equation, the Schrödinger equation, the Laplace equation and the CauchyRiemann equations.
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