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Title:  Dispersive estimates for the Schrödinger equation 
Author(s):  Green, William R. 
Director of Research:  Erdogan, M. Burak 
Doctoral Committee Chair(s):  Laugesen, Richard S. 
Doctoral Committee Member(s):  Erdogan, M. Burak; Hundertmark, Dirk; Bronski, Jared C. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Schrodinger operators
Dispersive estimates Oscillatory integrals 
Abstract:  In this document we explore the issue of $L^1\to L^\infty$ estimates for the solution operator of the linear Schr\"{o}dinger equation, \begin{align*} iu_t\Delta u+Vu&=0 &u(x,0)=f(x)\in \mathcal S(\R^n). \end{align*} We focus particularly on the five and seven dimensional cases. We prove that the solution operator precomposed with projection onto the absolutely continuous spectrum of $H=\Delta+V$ satisfies the following estimate $\e^{itH} P_{ac}(H)\_{L^1\to L^\infty} \lesssim t^{\frac{n}{2}}$ under certain conditions on the potential $V$. Specifically, we prove the dispersive estimate is satisfied with optimal assumptions on smoothness, that is $V\in C^{\frac{n3}{2}}(\R^n)$ for $n=5,7$ assuming that zero is regular, $V(x)\lesssim \langle x\rangle^{\beta}$ and $\nabla^j V(x)\lesssim \langle x\rangle^{\alpha}$, $1\leq j\leq \frac{n3}{2}$ for some $\beta>\frac{3n+5}{2}$ and $\alpha>3,8$ in dimensions five and seven respectively. We also show that for the five dimensional result one only needs that $V(x)\lesssim \langle x\rangle^{4}$ in addition to the assumptions on the derivative and regularity of the potential. This more than cuts in half the required decay rate in the first chapter. Finally we consider a problem involving the nonlinear Schr\"{o}dinger equation. In particular, we consider the following equation that arises in fiber optic communication systems, \begin{align*} iu_t+d(t) u_{xx}+u^2 u=0. \end{align*} We can reduce this to a nonlinear, nonlocal eigenvalue equation that describes the socalled dispersion management solitons. We prove that the dispersion management solitons decay exponentially in $x$ and in the Fourier transform of $x$. 
Issue Date:  20100831 
URI:  http://hdl.handle.net/2142/16998 
Rights Information:  Copyright 2010 William R. Green 
Date Available in IDEALS:  20100831 20120907 
Date Deposited:  201008 
This item appears in the following Collection(s)

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