Positive Encounters: An Investigative Study on the Effects of Interactions between Cations and Quantum Dots

Thesis (Ph.D.)--University of Washington, 2016-06

Some Inverse Problems in Analysis and Geometry

Thesis (Ph.D.)--University of Washington, 2016-08

Dispersive estimates for the Schrödinger equation

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 $V\in C^{\frac{n-3}{2}}(\R^n)$ 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 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 non-linear 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 non-linear, non-local eigenvalue equation that describes the so-called dispersion management solitons. We prove that the dispersion management solitons decay exponentially in $x$ and in the Fourier transform of $x$.

Discrete breathers in a two-dimensional hexagonal Fermi-Pasta-Ulam lattice

We consider a two-dimensional Fermi-Pasta-Ulam (FPU) lattice with hexagonal symmetry. Using asymptotic methods based on small amplitude ansatz, at third order we obtain a eduction to a cubic nonlinear Schr{\"o}dinger equation (NLS) for the breather envelope. However, this does not support stable soliton solutions, so we pursue a higher-order analysis yielding a generalised NLS, which includes known stabilising terms. We present numerical results which suggest that long-lived stationary and moving breathers are supported by the lattice. We find breather solutions which move in an arbitrary direction, an ellipticity criterion for the wavenumbers of the carrier wave, symptotic estimates for the breather energy, and a minimum threshold energy below which breathers cannot be found. This energy threshold is maximised for stationary breathers, and becomes vanishingly small near the boundary of the elliptic domain where breathers attain a maximum speed. Several of the results obtained are similar to those obtained for the square FPU lattice (Butt \& Wattis, {\em J Phys A}, {\bf 39}, 4955, (2006)), though we find that the square and hexagonal lattices exhibit different properties in regard to the generation of harmonics, and the isotropy of the generalised NLS equation.

Asymptotic analysis of combined breather-kink modes in a Fermi-Pasta-Ulam chain

We find approximations to travelling breather solutions of the one-dimensional Fermi-Pasta-Ulam (FPU) lattice. Both bright breather and dark breather solutions are found. We find that the existence of localised (bright) solutions depends upon the coefficients of cubic and quartic terms of the potential energy, generalising an earlier inequality derived by James [CR Acad Sci Paris 332, 581, (2001)]. We use the method of multiple scales to reduce the equations of motion for the lattice to a nonlinear Schr{\"o}dinger equation at leading order and hence construct an asymptotic form for the breather. We show that in the absence of a cubic potential energy term, the lattice supports combined breathing-kink waveforms. The amplitude of breathing-kinks can be arbitrarily small, as opposed to traditional monotone kinks, which have a nonzero minimum amplitude in such systems. We also present numerical simulations of the lattice, verifying the shape and velocity of the travelling waveforms, and confirming the long-lived nature of all such modes.

Gli integrali di cammino

Normalmente la meccanica quantistica non relativistica è ricavata a partire dal fatto che una particella al tempo t non può essere descritta da una posizione $x$ definita, ma piuttosto è descritta da una funzione, chiamata funzione d'onda, per cui vale l'equazione differenziale di Schr\"odinger, e il cui modulo quadro in $x$ viene interpretato come la probabilità di rilevare la particella in tale posizione. Quindi grazie all'equazione di Schr\"odinger si studia la dinamica della funzione d'onda, la sua evoluzione temporale. Seguendo quest'approccio bisogna quindi abbandonare il concetto classico di traiettoria di una particella, piuttosto quello che si studia è la "traiettoria" della funzione d'onda nei vari casi di campi di forze che agiscono sulla particella. In questa tesi si è invece scelto di studiare un approccio diverso, ma anch'esso efficace nel descrivere i fenomeni della meccanica quantistica non relativistica, formulato per la prima volta negli anni '50 del secolo scorso dal dott. Richard P. Feynman. Tale approccio consiste nel considerare una particella rilevata in posizione $x_a$ nell'istante $t_a$, e studiarne la probabilità che questa ha, nelle varie configurazioni dei campi di forze in azione, di giungere alla posizione $x_b$ ad un successivo istante $t_b$. Per farlo si associa ad ogni percorso che congiunge questi due punti spazio-temporali $a$ e $b$ una quantità chiamata ampiezza di probabilità del percorso, e si sviluppa una tecnica che permette di sommare le ampiezze relative a tutti gli infiniti cammini possibili che portano da $a$ a $b$, ovvero si integra su tutte le traiettorie $x(t)$, questo tipo di integrale viene chiamato integrale di cammino o più comunemente path integral. Il modulo quadro di tale quantità darà la probabilità che la particella rilevata in $a$ verrà poi rilevata in $b$.