Made available in DSpace on 2014-10-09T12:50:28Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:02:42Z (GMT). No. of bitstreams: 1 00768.pdf: 4150056 bytes, checksum: 8e8c0f1f3b65b77306c6aeb1ee8f3508 (MD5) / Dissertacao (Mestrado) / IEA/D / Instituto de Fisica, Universidade de Sao Paulo - IF/USP

Orientador: Alvaro de Souza Dutra / Banca: Denis Dalmazi / Banca: Marcelo Batista Hotti / Banca: Alexandre Grezzi de Miranda Schmidt / Banca: Elso Drigo Filho / Resumo: Nesta tese procuramos veri car e aprofundar os limites de validade dos chamados sistemas quânticos com simetria PT. Nestes tem-se, por exemplo, sistemas cuja hamiltoniana é não-hermitiana mas apresenta um espectro de energia real. Tal característica é usualmente justi cada pela presença da simetria PT (paridade e inversão temporal), muito embora não haja ainda uma demonstração bem aceita na literatutra desta propriedade de tais sistemas. Inicialmente estudamos sistemas quânticos não-relativísticos dependentes do tempo, sistemas em mais dimensões espaciais, a m de veri car possíveis limites da simetria PT na garantia da realidade do espectro. Logo depois estudamos sistemas quânticos relativísticos em 1+1D que possuem simetria PT com uma mistura adequada de potenciais: vetor, escalar e pseudo-escalar, sendo o potencial vetor complexo. Em seguida trabalhamos com densidades de lagrangiana com potenciais não-hermitianos em 1+1 dimensões espaço-temporais e em dimensões mais altas. A vantagem das baixas dimensões é que alguns sistemas possuem soluções não-perturbativas exatas. Finalmente, mostramos que não somente é possível ter um modelo consistente com dois campos escalares, mas também que a introdução de um número maior de campos permite que a densidade de energia também permaneça real. / Abstract: In this thesis we verify and try to deepen the limits of validity of the so called quantum systems with PT-symmetry. These are systems whose Hamiltonians are non-Hermitian but present real energy spectra. Such characteristic usually is justi ed by the presence of PT symmetry (parity and time inversion), despite of the fact that there is no well accepted demonstration in literature of this property of such systems yet. Initially we study timedependent non-relativistic quantum systems in one spatial dimension in order to verify possible limits for which the PT symmetry grants the reality of the spectra. Soon later we study relativistic quantum systems in 1+1D that they possess symmetry PT with an convenient mixing of complex vector plus scalar plus pseudoscalar potentials is considered. After that, we work with a Lagrangian density with such features in 1+1 space-time dimensions and higher dimensions, in the context of eld theory. The advantage of working in low dimensions is that, in such dimensions, some systems possess exact nonperturbative solutions. Finally, we show that not only it is possible to have a consistent model with two scalar elds, but also that the introduction of a bigger number of elds allows that the energy density also remains real. / Doutor

Made available in DSpace on 2014-10-09T12:50:28Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:02:42Z (GMT). No. of bitstreams: 1 00768.pdf: 4150056 bytes, checksum: 8e8c0f1f3b65b77306c6aeb1ee8f3508 (MD5) / Dissertacao (Mestrado) / IEA/D / Instituto de Fisica, Universidade de Sao Paulo - IF/USP

This thesis is concerned with analysis of spectral and orbital stability of solitary wave solutions to discrete and continuous PT-symmetric nonlinear Schroedinger equations. The main tools of this analysis are inspired by Hamiltonian systems, where conserved quantities can be used for proving orbital stability and Krein signature can be computed for prediction of instabilities in the spectrum of linearization. The main results are obtained for the chain of coupled pendula represented by a discrete NLS model, and for the trapped atomic gas represented by a continuous NLS model. Analytical results are illustrated with various numerical examples. / Thesis / Doctor of Philosophy (PhD)

The present thesis is devoted to the investigation of certain aspects of the large time behavior of the solutions of two nonlinear Schrödinger equations in dimension three in some suitable perturbative regimes. The first model consist in a Schrödinger equation with a concentrated nonlinearity obtained considering a {point} (or contact) interaction with strength $alpha$, which consists of a singular perturbation of the Laplacian described by a self adjoint operator $H_{alpha}$, and letting the strength $alpha$ depend on the wave function: $ifrac{du}{dt}= H_alpha u$, $alpha=alpha(u)$.It is well-known that the elements of the domain of a point interaction in three dimensions can be written as the sum of a regular function and a function that exhibits a singularity proportional to $|x - x_0|^{-1}$, where $x_0$is the location of the point interaction. If $q$ is the so-called charge of the domain element $u$, i.e. the coefficient of itssingular part, then, in order to introduce a nonlinearity, we let the strength $alpha$ depend on $u$ according to the law $alpha=-nu|q|^sigma$, with $nu > 0$. This characterizes the model as a focusing NLS with concentrated nonlinearity of power type. In particular, we study orbital and asymptotic stability of standing waves for such a model. We prove the existence of standing waves of the form $u (t)=e^{iomega t}Phi_{omega}$, which are orbitally stable in the range $sigma in (0,1)$, and orbitally unstable for $sigma geq 1.$ Moreover, we show that for $sigma in(0,frac{1}{sqrt 2}) cup left(frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right)$ every standing wave is asymptotically stable, in the following sense. Choosing an initial data close to the stationary state in the energy norm, and belonging to a natural weighted $L^p$ space which allows dispersive stimates, the following resolution holds: $u(t) =e^{iomega_{infty} t +il(t)} Phi_{omega_{infty}}+U_t*psi_{infty} +r_{infty}$, where $U_t$ is the free Schrödinger propagator,$omega_{infty} > 0$ and $psi_{infty}$, $r_{infty} inL^2(R^3)$ with $| r_{infty} |_{L^2} = O(t^{-p}) quadtextrm{as} ;; t right arrow +infty$, $p = frac{5}{4}$,$frac{1}{4}$ depending on $sigma in (0, 1/sqrt{2})$, $sigma in (1/sqrt{2}, 1)$, respectively, and finally $l(t)$ is a logarithmic increasing function that appears when $sigma in (frac{1}{sqrt{2}},sigma^*)$, for a certain $sigma^* in left(frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right]$. Notice that in the present model the admitted nonlinearities for which asymptotic stability of solitons is proved, are subcritical in the sense that it does not give rise to blow up, regardless of the chosen initial data. The second model is the energy critical focusing nonlinear Schrödinger equation $i frac{du}{dt}=-Delta u-|u|^4 u$. In this case we prove, for any $nu$ and $alpha_0$ sufficiently small, the existence of radial finite energy solutions of the form$u(t,x)=e^{ialpha(t)}lambda^{1/2}(t)W(lambda(t)x)+e^{iDeltat}zeta^*+o_{dot H^1} (1)$ as $tright arrow +infty$, where$alpha(t)=alpha_0ln t$, $lambda(t)=t^{nu}$,$W(x)=(1+frac13|x|^2)^{-1/2}$ is the ground state and $zeta^*$is arbitrarily small in $dot H^1$

Cette thèse est consacrée à l'étude de certains aspects du comportement en temps longs des solutions de deux équations de Schrödinger non-linéaires en dimension trois dans des régimes perturbatives convenables. Le premier modèle consiste en une équation de Schrödinger avec une non-linéarité concentrée obtenue en considérant une interaction ponctuelle de force $alpha$, c'est-à-dire une perturbation singulière du Laplacien décrite par un opérateur autoadjoint $H_{alpha}$, où la force $alpha$ dépend de la fonction d'onde : $ifrac{du}{dt}= H_alpha u$, $alpha=alpha(u)$. Il est bien connu que les éléments du domaine d'une interaction ponctuelle en trois dimensions peuvent être décrits comme la somme d'une fonction régulière et d'une fonction ayant une singularité proportionnelle à $|x - x_0|^{-1}$, où $x_0$ est l'emplacement du point d'interaction. Si $q$ est la charge d'un élément du domaine $u$, c'est-à-dire le coefficient de sa partie singulière, alors pour introduire une non-linéarité, on fait dépendre la force $alpha$ de $u$ selon la loi $alpha=-nu|q|^sigma$, avec $nu > 0$. Ce modèle est défini comme une équation de Schrödinger non-linéaire focalisant de type puissance avec une non-linéarité concentrée en $x_0$. Notre étude regarde la stabilité orbitale et asymptotique des ondes stationnaires de ce modèle. Nous prouvons l'existence d'ondes stationnaires de la forme $u (t)=e^{iomega t}Phi_{omega}$, qui soient orbitalement stables pour $sigma in (0,1)$ et orbitalement instables quand $sigma geq 1.$ De plus nous montrons que si $sigma in (0,frac{1}{sqrt 2}) cup (frac{1}{sqrt 2}, 1)$, alors chaque onde stationnaire est asymptotiquement stable, à savoir que pour des données initiales proches d'un état stationnaire dans la norme d'énergie et appartenant à un espace $L^p$ pondéré où les estimations dispersives sont valides, l'affirmation suivante est vérifiée : il existe $omega_{infty} > 0$ et $psi_{infty} in L^2(R^3)$ tel que $psi_{infty} = O_{L^2}(t^{-p})$ quand $t rightarrow +infty$, tel que $u(t) = e^{iomega_{infty} t +il(t)} Phi_{omega_{infty}} +U_t*psi_{infty} +r_{infty}$, où $U_t$ est le propagateur de Schrödinger libre, $p = frac{5}{4}$, $frac{1}{4}$ respectivement en fonction de $sigma in (0, 1/sqrt{2})$, $sigma in left( frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right)$, et $l(t)$ est une fonction à croissance logarithmique qui apparaît quand $sigma in (frac{1}{sqrt{2}}, sigma^*)$, où $sigma^* in left( frac{1}{sqrt{2}},frac{sqrt{3} +1}{2sqrt{2}} right]$. Notons que dans ce modèle les non-linéarités pour lesquelles on a la stabilité asymptotique sont sous-critiques dans le sens où quelle que soit la donnée initiale il n'y a pas de solutions explosives. Quant au deuxième modèle, il s'agit de l'équation de Schrödinger non-linéaire focalisant à énergie critique : $i frac{du}{dt}=-Delta u-|u|^4 u$. Pour ce cas, nous prouvons, pour tout $nu$ et $alpha_0$ suffisamment petits, l'existence de solutions radiales à énergie finie de la forme $u(t,x)=e^{ialpha(t)}lambda^{1/2}(t)W(lambda(t)x)+e^{iDelta t}zeta^*+o_{dot H^1} (1)$ tout $trightarrow +infty$, où $alpha(t)=alpha_0ln t$, $lambda(t)=t^{nu}$, $W(x)=(1+frac13|x|^2)^{-1/2}$ est l'état stationnaire et $zeta^*$ est arbitrairement petit en $dot H^1$ / The present thesis is devoted to the investigation of certain aspects of the large time behavior of the solutions of two nonlinear Schrödinger equations in dimension three in some suitable perturbative regimes. The first model consist in a Schrödinger equation with a concentrated nonlinearity obtained considering a {point} (or contact) interaction with strength $alpha$, which consists of a singular perturbation of the Laplacian described by a self adjoint operator $H_{alpha}$, and letting the strength $alpha$ depend on the wave function: $ifrac{du}{dt}= H_alpha u$, $alpha=alpha(u)$.It is well-known that the elements of the domain of a point interaction in three dimensions can be written as the sum of a regular function and a function that exhibits a singularity proportional to $|x - x_0|^{-1}$, where $x_0$is the location of the point interaction. If $q$ is the so-called charge of the domain element $u$, i.e. the coefficient of itssingular part, then, in order to introduce a nonlinearity, we let the strength $alpha$ depend on $u$ according to the law $alpha=-nu|q|^sigma$, with $nu > 0$. This characterizes the model as a focusing NLS with concentrated nonlinearity of power type. In particular, we study orbital and asymptotic stability of standing waves for such a model. We prove the existence of standing waves of the form $u (t)=e^{iomega t}Phi_{omega}$, which are orbitally stable in the range $sigma in (0,1)$, and orbitally unstable for $sigma geq 1.$ Moreover, we show that for $sigma in(0,frac{1}{sqrt 2}) cup left(frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right)$ every standing wave is asymptotically stable, in the following sense. Choosing an initial data close to the stationary state in the energy norm, and belonging to a natural weighted $L^p$ space which allows dispersive stimates, the following resolution holds: $u(t) =e^{iomega_{infty} t +il(t)} Phi_{omega_{infty}}+U_t*psi_{infty} +r_{infty}$, where $U_t$ is the free Schrödinger propagator,$omega_{infty} > 0$ and $psi_{infty}$, $r_{infty} inL^2(R^3)$ with $| r_{infty} |_{L^2} = O(t^{-p}) quadtextrm{as} ;; t right arrow +infty$, $p = frac{5}{4}$,$frac{1}{4}$ depending on $sigma in (0, 1/sqrt{2})$, $sigma in (1/sqrt{2}, 1)$, respectively, and finally $l(t)$ is a logarithmic increasing function that appears when $sigma in (frac{1}{sqrt{2}},sigma^*)$, for a certain $sigma^* in left(frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right]$. Notice that in the present model the admitted nonlinearities for which asymptotic stability of solitons is proved, are subcritical in the sense that it does not give rise to blow up, regardless of the chosen initial data. The second model is the energy critical focusing nonlinear Schrödinger equation $i frac{du}{dt}=-Delta u-|u|^4 u$. In this case we prove, for any $nu$ and $alpha_0$ sufficiently small, the existence of radial finite energy solutions of the form$u(t,x)=e^{ialpha(t)}lambda^{1/2}(t)W(lambda(t)x)+e^{iDeltat}zeta^*+o_{dot H^1} (1)$ as $tright arrow +infty$, where$alpha(t)=alpha_0ln t$, $lambda(t)=t^{nu}$,$W(x)=(1+frac13|x|^2)^{-1/2}$ is the ground state and $zeta^*$is arbitrarily small in $dot H^1$

Quantum Monte Carlo (QMC) methods are a class of stochastic techniques that can be used to compute the properties of electronic systems accurately from first principles. This thesis is mainly concerned with the development of trial wave functions for QMC. An extension of the backflow transformation to inhomogeneous electronic systems is presented and applied to atoms, molecules and extended systems. The backflow transformation I have developed typically retrieves an additional 50% of the remaining correlation energy at the variational Monte Carlo level, and 30% at the diffusion Monte Carlo level; the number of parameters required to achieve a given fraction of the correlation energy does not appear to increase with system size. The expense incurred by the use of backflow transformations is investigated, and it is found to scale favourably with system size. Additionally, I propose a single wave function form for studying the electron-hole system which includes pairing effects and is capable of describing all of the relevant phases of this system. The effectiveness of this general wave function is demonstrated by applying it to a particular transition between two phases of the symmetric electron-hole bilayer, and it is found that using a single wave function form gives a more accurate physical description of the system than using a different wave function to describe each phase. Both of these developments are new, and they provide a powerful set of tools for designing accurate wave functions. Backflow transformations are particularly important for systems with repulsive interactions, while pairing wave functions are important for attractive interactions. It is possible to combine backflow and pairing to further increase the accuracy of the wave function. The wave function technology that I have developed should therefore be useful across a very wide range of problems.

No description available.