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Title:  Global dynamics of Schrodinger and Dirac equations 
Author(s):  Toprak, Ebru 
Director of Research:  Erdogan, M. Burak 
Doctoral Committee Chair(s):  Tzirakis, Nikolaos 
Doctoral Committee Member(s):  Junge, Marius; Bronski, Jared 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Schrodinger equation, Dirac equation, dispersive estimate, thresholdenergy obstruction 
Abstract:  In this document, we study the linear Schr\"odinger operator and linear massive Dirac operator in the $L^1\to L^\infty$ settings. In Chapter~I, we focus on the two dimensional Schr\"odinger operator in the weighted $L^1(\R^2) \rightarrow L^{\infty}(\R^2)$ setting when there is a resonance of the first kind at zero energy. In particular, we show that if $V(x)\les \la x \ra ^{4}$ and there is only swave resonance at zero of $H$, then $$ \big\ w^{1} \big( e^{itH}P_{ac} f  {\f 1 {\pi it} } F f \big) \big\ _{\infty} \leq \frac {C} {t (\logt)^2 } \wf\_1,\,\,\,\,\,\,t>2,$$ with $w(x)=\log^2(2+x)$. Here $Ff={\f 14} \psi\la \psi,f \ra$, where $\psi$ is an swave resonance function. We also extend this result to matrix Schr\"odinger equations with potentials under similar conditions. In Chapter~II, we focus on the two and three dimensional massive Dirac equation with a potential. In two dimension, we show that the $t^{1}$ decay rate holds if the threshold energies are regular or if there are swave resonances at the threshold. We further show that, if the threshold energies are regular then a faster decay rate of $t^{1}(\log t)^{2}$ is attained for large $t$, at the cost of logarithmic spatial weights, which is not the case for the free Dirac equation. In three dimension, we show that the solution operator is composed of a finite rank operator that decays at the rate $t^{1/2}$ plus a term that decays at the rate $t^{3/2}$. 
Issue Date:  20180711 
Type:  Text 
URI:  http://hdl.handle.net/2142/101665 
Rights Information:  Copyright 2018, Ebru Toprak 
Date Available in IDEALS:  20180927 20200928 
Date Deposited:  201808 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois