In this document we explore the issue of $L^1\to L^\infty$ estimates for the solution operatorof 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 solutionoperator 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 $V\in C^{\frac{n-3}{2}}(\R^n)$ for$n=5,7$ assuming that zero is regular, $|V(x)|\lesssim \langle x\rangle^{-\beta}$ and $|\nabla^j V(x)|\lesssim \langle x\rangle^{-\alpha}$,$1\leq j\leq \frac{n-3}{2}$ for some $\beta>\frac{3n+5}{2}$ and $\alpha>3,8$ in dimensions five and sevenrespectively.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 thepotential.This more than cuts in half the required decay rate in the first chapter.Finally we consider a problem involving the non-linear Schr\"{o}dingerequation.In particular, we consider the following equation that arises in fiber optic communicationsystems,\begin{align*}iu_t+d(t) u_{xx}+|u|^2 u=0.\end{align*}We can reduce this to a non-linear, non-local eigenvalue equation that describes the so-called dispersionmanagement solitons.We prove that the dispersion management solitons decay exponentially in $x$ and in the Fourier transform of $x$.
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