Ce travail est constitué de deux parties indépendantes traitant chacune d'un problème issu de la théorie du contrôle des équations aux dérivées partielles. La première partie est consacrée à l'étude d'un feedback explicite et déjà connu, s'appliquant à des systèmes linéaires, réversibles en temps et éventuellement munis d'un opérateur de contrôle non-borné. On justifie le caractère bien posé du problème en boucle fermée via la théorie des semi-groupes puis on étudie le taux de décroissance des solutions du système régulé. La seconde partie concerne un problème d'observation pour la corde vibrante : on détermine comment choisir des instants d'observation pour que la position de la corde à ces instants permette de retrouver les conditions initiales tout en préservant une certaine régularité. La méthode, qui repose sur des résultats d'approximation diophantienne, est ensuite étendue à d'autres systèmes. En utilisant une méthode de dualité on démontre aussi un résultat de contrôlabilité exacte. / This works contains two independent parts, each one dealing with the control of partial differential equations. In the first part, we study an explicit and already known feedback law that applies to linear, time-reversible systems, with a possibly unbounded control operator. We prove the well-posedness of the closed-loop problem in the semi-group framework and we study the decay rate of the solutions. In the second part, we give conditions on the choice of some time instants, such that the positions of a vibrating string (or beam) at these times enable to recover the initial data. The method relies on Diophantine approximation results. Using a duality method, we give a related exact controllability result.

[pt] Consideramos o oscilador harmônico com amortecimento aleatório em presença de ruído externo. Os ruídos, representando perturbações externas e internas, são modelados pelo processo de Ornstein-Uhlenbeck ou ruído branco e pelo processo dicotômico ou ruído branco, respectivamente. Usando técnicas de sistemas dinâmicos, analisamos o valor médio e a dispersão da posição e da velocidade do oscilador harmônico estocástico, apresentando resultados analíticos e numéricos. Em particular, obtemos expressões para a expansão de baixa-ordem em relação ao tempo de correlação da perturbação interna, no caso da atuação do ruído dicotômico. Finalmente, usando o modelo de oscilador harmônico com amortecimento aleatório como referência, investigamos a série intradiária de preços do mercado brasileiro. / [en] We consider the random damping harmonic oscillator in presence of external noise. The noises, representing external and internal perturbations, are modeled as an Ornstein-Uhlenbeck process or a white noise and as a dichotomous process or a white noise, respectively. Using dynamical systems tools, we analyze the expected value as well as the dispersion of the stochastic harmonic oscillator s position and velocity, presenting analytical and numerical results. In particular, we also provide expressions for the low-order expansion in the correlation time of the internal perturbation, in the case the dichotomous noise is at play. Using random damped harmonic oscillator model as a reference, we conclude by investigating the intra-day Brazilian stock price series.

Normal shock problems in high enthalpy flows are of special interests to aerodynamicists and fluid dynamicists. When the shock Mach number become hypersonic and increasing further, the gas passing through the shock is compressed resulting in increase in temperature and pressure. As the Mach number increases the internal degrees of freedom of the diatomic molecules are activated to an increasing extent when it crosses the shock resulting dissociation especially for high enthalpy flows. Hence dissociation of diatomic molecules must be taken into account in the determination of some of the aerodynamic parameters. This thermal and chemical process can be divided into three types such as nearly frozen, non-equilibrium and nearly non-equilibrium depending on the rates of reaction and excitation. For typical re-entry conditions of spacecrafts into a planets atmosphere, dissociation reactions of the molecules is dominant in the stagnation flow. Further in the stagnation region of the flow field one of the most important parameter that characterizes the flow field is the shock stand-off distance. This parameter is often employed for validation purposes of numerical methods as well as for non-reactive and reactive gases. For high Mach number flows the shock is very close to the body hence experimental determination of shock stand-off distance is very difficult and there would be relatively large errors. Therefore the theoretical determination of this parameter is of great significance in the discussion of this physical phenomenon. There are some works which presents how the dissociation behind shock affects the shock stand-off distance. Thus the dissociation behind the shock is a very important process which has great impact in aerodynamic flight and design. In this present work we studied how dissociation of diatoms occur behind a normal shock. Treanor and Marrone (1962) proposed CVD(coupled vibration-dissociation) model for diatoms by assuming diatom as a harmonic oscillator with a cut-off level. But actually diatoms are not harmonic oscillator, because spectroscopic data of energy level spacing is not like harmonic oscillator. For this reason, Treanor, Rich, and Rehm(1968) used anharmonic oscillator model for diatoms to study vibrational relaxation. Taking the anharmonicity of diatom, Philip Morse(1929) gave a formula for potential energy levels for diatoms, which is known to express the experimental values quite accurately. Unlike the energy levels of the harmonic oscillator potential, which are evenly spaced , the Morse potential level spacing decreases as the energy approaches the dissociation energy and then it is continuous. So it is quite accurate to take Morse oscillator theory for diatomic dissociation instead of harmonic oscillator with a cut-off level. We have used Morse oscillator theory to derive a dissociation-recombination reaction rate equation for diatom. To derive the rate equation we have used the transition probability between different vibrational energy levels . The rate equation is numerically solved to get the different flow variables behind the shock. The result of the present work has been compared with some of the previous work. Some of the flow variables are well matching with the previous work and some has discrepancy near the shock but well matching after few distance from the shock. We have also studied under what conditions the post shock flow shows self-similar behavior in its scaling relations. It is shown that as far as there is no dissociation, we could expect to obtain self-similar solutions. However, when there is dissociation, the non-equillibrium nature of the phenomenon disrupts the self-similar nature of the flow.

In this paper we derive the eigenfunctions to the Hamiltonian operator associated with the Harmonic Oscillator, and show that they are given by the Hermite functions. Then we prove that the Hermite functions form an orthonormal basis in the underlying Hilbert space. We also classify the inverse to the Hamiltonian operator as a Schatten-von Neumann operator. Finally, we derive the fundamental solution to the Schrödinger Equation corresponding to the Harmonic Oscillator using Mehler’s formula.

In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.

[pt] A Mecânica Quântica é uma das teorias mais bem sucedidas de todos os tempos. Não apenas tem um grande poder preditivo, como também mudou completamente a maneira como entendemos a física. No entanto, a Mecânica Quântica não prevê o próprio alcance, e, em princípio, a descrição probabilística deveria ser válida em nosso mundo macroscópico. Mas isso não acontece. Um ponto central do porquê a descrição quântica é substituída pela Mecânica Clássica, é a descoerência. A interação dos muitos graus de liberdade de um ambiente macroscópico faz com que seja extremamente difícil medirmos as propriadades quânticas de um sistema. Nesse contexto, exploramos como ainda podemos detectar efeitos não-clássicos de osciladores harmônicos em um regime intermediário, através da optomecânica. Neste trabalho apresentamos fundamentos do formalismo da optomecânica, tanto a dinâmica unitária, quanto para sistemas quânticos abertos. Depois, discutimos dois sistemas optomecânicos distintos, ressaltando como podemos investigar a presença de características quânticas. Além disso, discutiremos outras abordagens para identifcar características quânticas de osciladores harmônicos em situações cada vez mais próximas ao regime macroscópico. / [en] Quantum Mechanics is one of the most successful theories of all time. It not only has a great predictive power, but also has completely changed the way physics is understood. However, Quantum Mechanics does not predict its own range of validity, and, in principle, the probabilistic description should be valid in our macroscopic world. But it does not happen. In the core of the explanation of why the description from Quantum Mechanics is substituted by Classical Mechanics lies decoherence. The interaction of the many unseen degrees of freedom from a macroscopic environment with a quantum system makes it extremely hard to measure its quantum properties. In this context, we explore how one can still detect non-classicality of oscillators in a intermediate regime, whether in a mesoscopic scale or a oscillator with a macroscopic number of excitations, via an optomechanical description. In this work, we present the basics of the formalism of optomechanics, both in the unitary dynamics and in an open quantum system approach. We then discuss two different optomechanical systems, highlighting how we can perceive its quantum features. At last, we discuss other possible schemes to identify the quantum nature of harmonic oscillators in situations of increasing macroscopic nature.

L'objectif de ce mémoire est de démontrer certaines propriétés géométriques des fonctions propres de l'oscillateur harmonique quantique. Nous étudierons les domaines nodaux, c'est-à-dire les composantes connexes du complément de l'ensemble nodal. Supposons que les valeurs propres ont été ordonnées en ordre croissant. Selon un théorème fondamental dû à Courant, une fonction propre associée à la $n$-ième valeur propre ne peut avoir plus de $n$ domaines nodaux. Ce résultat a été prouvé initialement pour le laplacien de Dirichlet sur un domaine borné mais il est aussi vrai pour l'oscillateur harmonique quantique isotrope. Le théorème a été amélioré par Pleijel en 1956 pour le laplacien de Dirichlet. En effet, on peut donner un résultat asymptotique plus fort pour le nombre de domaines nodaux lorsque les valeurs propres tendent vers l'infini. Dans ce mémoire, nous prouvons un résultat du même type pour l'oscillateur harmonique quantique isotrope. Pour ce faire, nous utiliserons une combinaison d'outils classiques de la géométrie spectrale (dont certains ont été utilisés dans la preuve originale de Pleijel) et de plusieurs nouvelles idées, notamment l'application de certaines techniques tirées de la géométrie algébrique et l'étude des domaines nodaux non-bornés. / The aim of this thesis is to explore the geometric properties of eigenfunctions of the isotropic quantum harmonic oscillator. We focus on studying the nodal domains, which are the connected components of the complement of the nodal (i.e. zero) set of an eigenfunction. Assume that the eigenvalues are listed in an increasing order. According to a fundamental theorem due to Courant, an eigenfunction corresponding to the $n$-th eigenvalue has at most $n$ nodal domains. This result has been originally proved for the Dirichlet eigenvalue problem on a bounded Euclidean domain, but it also holds for the eigenfunctions of a quantum harmonic oscillator. Courant's theorem was refined by Pleijel in 1956, who proved a more precise result on the asymptotic behaviour of the number of nodal domains of the Dirichlet eigenfunctions on bounded domains as the eigenvalues tend to infinity. In the thesis we prove a similar result in the case of the isotropic quantum harmonic oscillator. To do so, we use a combination of classical tools from spectral geometry (some of which were used in Pleijel’s original argument) with a number of new ideas, which include applications of techniques from algebraic geometry and the study of unbounded nodal domains.

A dissertação pode ser dividida em duas partes: a primeira contém uma adaptação do modelo fenomenológico de Einstein conhecido como \"método dos coeficientes A e B\". As modificações são feitas no contexto da Eletrodinâmica Estocástica, uma teoria na qual as flutuações de ponto zero do campo eletromagnético são consideradas reais e clássicas. Nós obtemos, num estudo não relativístico e clássico, relações entre a energia e momento de partículas livres e a frequência da radiação transferida. Estas relações coincidem com as bem conhecidas relações que representam a conservação do quadrivetor momento-energia em espalhamento fóton-elétron. Na segunda parte nós tentamos descrever, de uma maneira qualitativa, o efeito Compton no espírito da Eletrodinâmica Estocástica. Encontramos indicações de que a ação combinada da força da reação da radiação e das flutuações de ponto zero é capaz de conferir à partícula carregada uma alta velocidade de recuo, e verificamos que a mesma é justamente a necessária para explanar o deslocamento da frequência observado como sendo devido ao efeito Doppler. Também calculamos a seção de choque diferencial para o espalhamento de radiação e encontramos a mesma expressão obtida por Compton no seu trabalho fundamental de 1923. / The dissertation may be divided in two parts: the first one contains an adaptation of Einstein\'s phenomenological model known as \"method of coefficients A and B\". The modifications are done in the framework of Stochastic Electrodynamics, a theory in wich the zero point fluctuations of the electromagnetic field are considered real and classical. We obtain, in a nonrelativistic and Classical approach, relations among the energy and momentum of free particles and the frequency of the exchanged radiation. These relations are coincident with the well known ones who depict the four-momentum conservation in photon-electron scattering. In the second part, we try to describe, in a qualitative manner, the Compton scattering in the spirit of Stochastic Electrodynamics. We find indications that the combined action of the radiation reaction force and the zero point flutuating field are able to give the charged particle a high recoil velocity, and we verify this one is just the necessary to explain the frequency shift as due to a double Doppler shift. We also calculate the differential cross section for the radiation scattering and we find the same expression as obtained by Compton in his fundamental work of 1023.

The vacuum, as described by Quantum Field Theory, is not as empty as classical physics once led us to believe. In fact, it is characterised by an infinite energy stored in the ground state of its constituent fields. This infinite energy has real, tangible effects on the macroscopic clusters of matter that make up our universe. Moreover, the configuration of these clusters of matter within the vacuum in turn influences the form of the vacuum itself and so forth. In this work, we shall consider the changes to the quantum vacuum brought about by the presence of time-dependent dielectrics. Such changes are thought to be responsible for phenomena such as the simple and dynamical Casimir effects and Quantum Friction. After introducing the physical and mathematical descriptions of the electromagnetic quantum vacuum, we will begin by discussing some of the basic quasi-static effects that stem directly from the existence of an electromagnetic ground state energy, known as the \textit{zero-point energy}. These effects include the famous Hawking radiation and Unruh effect amongst others. We will then use a scenario similar to that which exhibits Cherenkov radiation in order to de-mystify the 'negative frequency' modes of light that often occur due to a Doppler shift in the presence of media moving at a constant velocity by showing that they are an artefact of the approximation of the degrees of freedom of matter to a macroscopic permittivity function. Here, absorption and dissipation of electromagnetic energy will be ignored for simplicity. The dynamics of an oscillator placed within this moving medium will then be considered and we will show that when the motion exceeds the speed of light in the dielectric, the oscillator will begin to absorb energy from the medium. It will be shown that this is due to the reversal of the 'radiation damping' present for lower velocity of stationary cases. We will then consider how the infinite vacuum energy changes in the vicinity, but outside, of this medium moving with a constant velocity and show that the presence of matter removes certain symmetries present in empty space leading to transfers of energy between moving bodies mediated by the electromagnetic field. Following on from this, we will then extend our considerations by including the dissipation and dispersion of electromagnetic energy within magneto-dielectrics by using a canonically quantised model referred to as 'Macroscopic QED'. We will analyse the change to the vacuum state of the electromagnetic field brought about by the presence of media with an arbitrary time dependence. It will be shown that this leads to the creation of particles tantamount to exciting the degrees of freedom of both the medium and the electromagnetic field. We will also consider the effect these time-dependencies have on the two point functions of the field amplitudes using the example of the electric field. Finally, we will begin the application of the macroscopic QED model to the path integral methods of quantum field theory with the purpose of making use of the full range of perturbative techniques that this entails, leaving the remainder of this adaptation for future work.

A dissertação pode ser dividida em duas partes: a primeira contém uma adaptação do modelo fenomenológico de Einstein conhecido como \"método dos coeficientes A e B\". As modificações são feitas no contexto da Eletrodinâmica Estocástica, uma teoria na qual as flutuações de ponto zero do campo eletromagnético são consideradas reais e clássicas. Nós obtemos, num estudo não relativístico e clássico, relações entre a energia e momento de partículas livres e a frequência da radiação transferida. Estas relações coincidem com as bem conhecidas relações que representam a conservação do quadrivetor momento-energia em espalhamento fóton-elétron. Na segunda parte nós tentamos descrever, de uma maneira qualitativa, o efeito Compton no espírito da Eletrodinâmica Estocástica. Encontramos indicações de que a ação combinada da força da reação da radiação e das flutuações de ponto zero é capaz de conferir à partícula carregada uma alta velocidade de recuo, e verificamos que a mesma é justamente a necessária para explanar o deslocamento da frequência observado como sendo devido ao efeito Doppler. Também calculamos a seção de choque diferencial para o espalhamento de radiação e encontramos a mesma expressão obtida por Compton no seu trabalho fundamental de 1923. / The dissertation may be divided in two parts: the first one contains an adaptation of Einstein\'s phenomenological model known as \"method of coefficients A and B\". The modifications are done in the framework of Stochastic Electrodynamics, a theory in wich the zero point fluctuations of the electromagnetic field are considered real and classical. We obtain, in a nonrelativistic and Classical approach, relations among the energy and momentum of free particles and the frequency of the exchanged radiation. These relations are coincident with the well known ones who depict the four-momentum conservation in photon-electron scattering. In the second part, we try to describe, in a qualitative manner, the Compton scattering in the spirit of Stochastic Electrodynamics. We find indications that the combined action of the radiation reaction force and the zero point flutuating field are able to give the charged particle a high recoil velocity, and we verify this one is just the necessary to explain the frequency shift as due to a double Doppler shift. We also calculate the differential cross section for the radiation scattering and we find the same expression as obtained by Compton in his fundamental work of 1023.