A two-dimensional integrodi erential equation resulting from the use of potential harmonics expansion in the many-body Schr odinger equation is used to study ground-state properties of selected few-body nuclear systems. The equation takes into account twobody correlations in the system and is applicable to few- and many-body systems. The formulation of the equation involves the use of the Jacobi coordinates to de ne relevant global coordinates as well as the elimination of center-of-mass dependence. The form of the equation does not depend on the size of the system. Therefore, only the interaction potential is required as input. Di erent nucleon-nucleon potentials and hyperon-nucleon potentials are employed to construct the Hamiltonian of the systems. The results obtained are in good agreement with those obtained using other methods. / Physics / M.Sc. (Physics)

We show that Einstein’s gravitational field equations for the Friedmann- Robertson-Lemaître-Walker (FRLW) and for two conformal versions of the Bianchi I and Bianchi V perfect fluid scalar field cosmological models, can be equivalently reformulated in terms of a single equation of either generalized Ermakov-Milne- Pinney (EMP) or (non)linear Schrödinger (NLS) type. This work generalizes or presents an alternative to similar reformulations published by the authors who inspired this thesis: R. Hawkins, J. Lidsey, T. Christodoulakis, T. Grammenos, C. Helias, P. Kevrekidis, G. Papadopoulos and F.Williams. In particular we cast much of these authors’ works into a single framework via straightforward derivations of the EMP and NLS equations from a simple linear combination of the relevant Einstein equations. By rewriting the resulting expression in terms of the volume expansion factor and performing a change of variables, we obtain an uncoupled EMP or NLS equation that is independent of the imposition of additional conservation equations. Since the correspondences shown here present an alternative route for obtaining exact solutions to Einstein’s equations, we reconstruct many known exact solutions via their EMP or NLS counterparts and show by numerical analysis the stability properties of many solutions.

<p>We consider existence and stability of breather solutions to discrete nonlinear Schrodinger (dNLS) and discrete Klein-Gordon (dKG) equations near the anti-continuum limit, the limit of the zero coupling constant. For sufficiently small coupling, discrete breathers can be uniquely extended from the anti-continuum limit where they consist of periodic oscillations on excited sites separated by "holes" (sites at rest).</p> <p>In the anti-continuum limit, the dNLS equation linearized about its discrete breather has a spectrum consisting of the zero eigenvalue of finite multiplicity and purely imaginary eigenvalues of infinite multiplicities. Splitting of the zero eigenvalue into stable and unstable eigenvalues near the anti-continuum limit was examined in the literature earlier. The eigenvalues of infinite multiplicity split into bands of continuous spectrum, which, as observed in numerical experiments, may in turn produce internal modes, additional eigenvalues on the imaginary axis. Using resolvent analysis and perturbation methods, we prove that no internal modes bifurcate from the continuous spectrum of the dNLS equation with small coupling.</p> <p>Linear stability of small-amplitude discrete breathers in the weakly-coupled KG lattice was considered in a number of papers. Most of these papers, however, do not consider stability of discrete breathers which have "holes" in the anti-continuum limit. We use perturbation methods for Floquet multipliers and analysis of tail-to-tail interactions between excited sites to develop a general criterion on linear stability of multi-site breathers in the KG lattice near the anti-continuum limit. Our criterion is not restricted to small-amplitude oscillations and it allows discrete breathers to have "holes" in the anti-continuum limit.</p> / Doctor of Philosophy (PhD)

This thesis is concerned with the inverse problem of determining a unitary connection $A$ on a Hermitian vector bundle $E$ of rank $m$ over a compact Riemannian manifold $(M, g)$ from the Dirichlet-to-Neumann (DN) map $\Lambda_A$ of the associated connection Laplacian $d_A^*d_A$. The connection is to be determined up to a unitary gauge equivalence equal to the identity at the boundary. In our first approach to the problem, we restrict our attention to conformally transversally anisotropic (cylindrical) manifolds $M \Subset \mathbb{R}\times M_0$. Our strategy can be described as follows: we construct the special Complex Geometric Optics solutions oscillating in the vertical direction, that concentrate near geodesics and use their density in an integral identity to reduce the problem to a suitable $X$-ray transform on $M_0$. The construction is based on our proof of existence of Gaussian Beams on $M_0$, which are a family of smooth approximate solutions to $d_A^*d_Au = 0$ depending on a parameter $\tau \in \mathbb{R}$, bounded in $L^2$ norm and concentrating in measure along geodesics when $\tau \to \infty$, whereas the small remainder (that makes the solution exact) can be shown to exist by using suitable Carleman estimates. In the case $m = 1$, we prove the recovery of the connection given the injectivity of the $X$-ray transform on $0$ and $1$-forms on $M_0$. For $m > 1$ and $M_0$ simple we reduce the problem to a certain two dimensional $\textit{new non-abelian ray transform}$. In our second approach, we assume that the connection $A$ is a $\textit{Yang-Mills connection}$ and no additional assumption on $M$. We construct a global gauge for $A$ (possibly singular at some points) that ties well with the DN map and in which the Yang-Mills equations become elliptic. By using the unique continuation property for elliptic systems and the fact that the singular set is suitably small, we are able to propagate the gauges globally. For the case $m = 1$ we are able to reconstruct the connection, whereas for $m > 1$ we are forced to make the technical assumption that $(M, g)$ is analytic in order to prove the recovery. Finally, in both approaches we are using the vital fact that is proved in this work: $\Lambda_A$ is a pseudodifferential operator of order $1$ acting on sections of $E|_{\partial M}$, whose full symbol determines the full Taylor expansion of $A$ at the boundary.

The multiconfiguration time-dependent Hartree-Fock (MCTDHF) and multiconfiguration time-dependent Hartree (MCTDH) methods are employed to investigate nonperturbative multielectron dynamics in finite quantum systems. MCTDHF is a powerful tool that allows for the investigation of multielectron dynamics in strongly perturbed quantum systems. We have developed an MCTDHF code that is capable of treating problems involving three dimensional (3D) atoms and molecules exposed to strong laser fields. This code will allow for the theoretical treatment of multielectron phenomena in attosecond science that were previously inaccessible. These problems include complex ionization processes in pump-probe experiments on noble gas atoms, the nonlinear effects that have been observed in Ne atoms in the presence of an x-ray free-electron laser (XFEL) and the molecular rearrangement of cations after ionization. An implementation of MCTDH that is optimized for two electrons, each moving in two dimensions (2D), is also presented. This implementation of MCTDH allows for the efficient treatment of 2D spin-free systems involving two electrons; however, it does not scale well to 3D or to systems containing more that two electrons. Both MCTDHF and MCTDH were used to treat 2D problems in nanophysics and attosecond science. MCTDHF is used to investigate plasmon dynamics and the quantum breathing mode for several electrons in finite lateral quantum dots. MCTDHF is also used to study the effects of manipulating the potential of a double lateral quantum dot containing two electrons; applications to quantum computing are discussed. MCTDH is used to examine a diatomic model molecular system exposed to a strong laser field; nonsequential double ionization and high harmonic generation are studied and new processes identified and explained. An implementation of MCTDHF is developed for nonuniform tensor product grids; this will allow for the full 3D implementation of MCTDHF and will provide a means to investigate a wide variety of problems that cannot be currently treated by any other method. Finally, the time it takes for an electron to tunnel from a bound state is investigated; a definition of the tunnel time is established and the Keldysh time is connected to the wavefunction dynamics.

The central objective of this thesis is to suggest and develop one simple, unified method for communication over optical fiber networks, valid for all values of dispersion and nonlinearity parameters, and for a single-user channel or a multiple-user network. The method is based on the nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees of freedom in such models, in much the same way that the Fourier transform does for linear systems. In this thesis, this observation is exploited for data transmission over integrable channels such as optical fibers, where pulse propagation is governed by the nonlinear Schr\"odinger (NLS) equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear spectrum of the signal. Just as the (ordinary) Fourier transform converts a linear convolutional channel into a number of parallel scalar channels, the nonlinear Fourier transform converts a nonlinear dispersive channel described by a \emph{Lax convolution} into a number of parallel scalar channels. Since, in the spectral coordinates the NLS equation is multiplicative, users of a network can operate in independent nonlinear frequency bands with no deterministic inter-channel interference. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This thesis lays the foundations of such a nonlinear frequency-division multiplexing system.

The central objective of this thesis is to suggest and develop one simple, unified method for communication over optical fiber networks, valid for all values of dispersion and nonlinearity parameters, and for a single-user channel or a multiple-user network. The method is based on the nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees of freedom in such models, in much the same way that the Fourier transform does for linear systems. In this thesis, this observation is exploited for data transmission over integrable channels such as optical fibers, where pulse propagation is governed by the nonlinear Schr\"odinger (NLS) equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear spectrum of the signal. Just as the (ordinary) Fourier transform converts a linear convolutional channel into a number of parallel scalar channels, the nonlinear Fourier transform converts a nonlinear dispersive channel described by a \emph{Lax convolution} into a number of parallel scalar channels. Since, in the spectral coordinates the NLS equation is multiplicative, users of a network can operate in independent nonlinear frequency bands with no deterministic inter-channel interference. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This thesis lays the foundations of such a nonlinear frequency-division multiplexing system.

CAPES / Neste estudo, os espectros Raman ressonante das moléculas de H2 e O2 foram computados através da resolução numérica da equação de Schrödinger vibracional utilizando potenciais completamente anarmônicos, obtidos através de métodos ab initio multiconfiguracionais. O problema vibracional foi resolvido através da simulação de Monte Carlo Quântico Variacional Modificado (MCQVM) e do método de Interação de Configurações Vibracional (VCI). As intensidades RR foram calculadas através da teoria independente do tempo do efeito RR. Exceto pela PES do estado eletrônico excitado B3Σ− u da molécula de O2, a qual não pôde ser plenamente descrita devido a um cruzamento evitado entre estados eletrônicos, as demais PESs apresentaram uma boa concordância com os valores experimentais das constantes espectroscópicas (Re, Te, De, Be, We e WeXe). Constatou-se através do desvio teórico-experimental que as energias vibracionais geradas via simulação de MCQVM são, de maneira geral, bastante acuradas para os estados vibracionais de mais baixa energia. Entretanto, à medida que necessitou-se otimizar um número elevado de estados vibracionais de energia superior, necessários para o cálculo da polarizabilidade RR, houve um acréscimo significativo no tempo computacional dos cálculos MCQVM, motivando a adoção do método VCI, no intuito de tentar reduzir o tempo de processamento e também conseguir resultados mais acurados. No que tange ao cálculo da seção de choque RR das moléculas de H2 e O2, constatou-se que a convergência das seções de choque, com relação ao número de estados vibracionais intermediários, é mais rápida quando o potencial de interação é harmônico. A adição da correção vibrônica de Herzberg-Teller (HT) foi bastante pronunciada para as duas moléculas avaliadas, e tendem a atenuar as intensidades das transições RR com relação aos valores obtidos com a aproximação de Franck-Condon. Constatou-se que a meialargura à meia-altura (Γ) do estado vibrônico excitado tem pouca influência sobre a magnitude das seções de choque RR dos sistemas estudados, podendo ser variado em mais de uma ordem de grandeza sem que isso afete substancialmente as intensidades RR. Por fim, verificou-se que a obtenção das intensidades RR por meio de funções de onda vibracionais anarmônicas demanda um número elevado de estados vibracionais intermediários (algumas dezenas) para que haja uma boa convergência nas seções de choque. Esse número é bastante superior ao observado nos cálculos harmônicos. / In this study, the resonance Raman spectra of H2 and O2 molecules were computed by the numerical resolution of the vibrational Schrödinger equation employing fully anharmonic potentials, obtained by Multiconfigurational ab initio methods. The vibrational problem was solved through the Modified Variational Quantum Monte Carlo (MVQMC) simulation and the Vibrational Configurations Interaction (VCI) methods. The time independent framework of the RR effect was used to compute the RR intensities. The Potential Energy Surfaces (PESs) were in good agreement with the experimental values of the spectroscopic constants (Re, Te, De, Be, We e WeXe), except for the excited electronic state B3Σ− u PES of the O2 molecule, which could not be fully described due to an avoided crossing between electronic states. It was verified through the theoretical-experimental deviation values that the vibrational energies generated by MVQMC simulations are, in general, very accurate for the lowest energy vibrational states. However, as it was necessary to optimize higher energy states, required in the calculation of the RR polarizability, the computational time of MVQMC method was greatly increased, motivating the use of VCI method in order to try to reduce processing time and achieve results that are more accurate. Regarding to calculating the RR cross sections of the H2 and O2 molecules, results have showed that the convergence is faster when using the harmonic potential instead of the anharmonic potential. The addition of vibronic coupling effects was quite impactful for both molecular systems evaluated, and tended to attenuate the RR transition intensities relative to the values obtained with the Franck–Condon contribution. It was verified that the bandwidth variation of the excited electronic state (Γ parameter) pose little influence in the RR cross sections of the addressed molecular systems, therefore being able to vary Γ in more than an order of magnitude without substantially affecting the RR intensities. Ultimately, the target property is largely influenced by the number of vibrational wave functions, requiring considerable quantities of intermediate vibrational states (dozens) for a good convergence of the anharmonic RR cross sections, which is a much larger quantity compared to the harmonic calculations.

The multiconfiguration time-dependent Hartree-Fock (MCTDHF) and multiconfiguration time-dependent Hartree (MCTDH) methods are employed to investigate nonperturbative multielectron dynamics in finite quantum systems. MCTDHF is a powerful tool that allows for the investigation of multielectron dynamics in strongly perturbed quantum systems. We have developed an MCTDHF code that is capable of treating problems involving three dimensional (3D) atoms and molecules exposed to strong laser fields. This code will allow for the theoretical treatment of multielectron phenomena in attosecond science that were previously inaccessible. These problems include complex ionization processes in pump-probe experiments on noble gas atoms, the nonlinear effects that have been observed in Ne atoms in the presence of an x-ray free-electron laser (XFEL) and the molecular rearrangement of cations after ionization. An implementation of MCTDH that is optimized for two electrons, each moving in two dimensions (2D), is also presented. This implementation of MCTDH allows for the efficient treatment of 2D spin-free systems involving two electrons; however, it does not scale well to 3D or to systems containing more that two electrons. Both MCTDHF and MCTDH were used to treat 2D problems in nanophysics and attosecond science. MCTDHF is used to investigate plasmon dynamics and the quantum breathing mode for several electrons in finite lateral quantum dots. MCTDHF is also used to study the effects of manipulating the potential of a double lateral quantum dot containing two electrons; applications to quantum computing are discussed. MCTDH is used to examine a diatomic model molecular system exposed to a strong laser field; nonsequential double ionization and high harmonic generation are studied and new processes identified and explained. An implementation of MCTDHF is developed for nonuniform tensor product grids; this will allow for the full 3D implementation of MCTDHF and will provide a means to investigate a wide variety of problems that cannot be currently treated by any other method. Finally, the time it takes for an electron to tunnel from a bound state is investigated; a definition of the tunnel time is established and the Keldysh time is connected to the wavefunction dynamics.

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