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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
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Schrodinger equation, Dirac equation, dispersive estimate, threshold-energy 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 s-wave resonance at zero of $H$, then $$ \big\| w^{-1} \big( e^{itH}P_{ac} f - {\f 1 {\pi it} } F f \big) \big\| _{\infty} \leq \frac {C} {|t| (\log|t|)^2 } \|wf\|_1,\,\,\,\,\,\,|t|>2,$$ with $w(x)=\log^2(2+|x|)$. Here $Ff=-{\f 14} \psi\la \psi,f \ra$, where $\psi$ is an s-wave 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 s-wave 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:2018-07-11
Rights Information:Copyright 2018, Ebru Toprak
Date Available in IDEALS:2018-09-27
Date Deposited:2018-08

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