[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.

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.

This thesis involves the study of four problems in the area of nonlinear vibrations, using the asymptotic method of multiple scales(MMS). Accordingly, it consists of four sequentially arranged parts. In the first part of this thesis we study some nonlinear dynamics related to the amplitude control of a lightly damped, resonantly forced, harmonic oscillator. The slow flow equations governing the evolution of amplitude and phase of the controlled system are derived using the MMS. Upon choice of a suitable control law, the dynamics is represented by three coupled ,nonlinear ordinary differential equations involving a scalar free parameter. Preliminary study of this system using the bifurcation analysis package MATCONT reveals the presence of Hopf bifurcations, pitchfork bifurcations, and limit cycles which seem to approach a homoclinic orbit. However, close approach to homoclinic orbit is not attained using MATCONT due to an inherent limitation of time domain-based continuation algorithms. To continue the limit cycles closer to the homoclinic point, a new algorithm is proposed. The proposed algorithm works in phase space with an ordered set of points on the limit cycle, along with spline interpolation. The algorithm incorporates variable stretching of arclength based on local curvature, through the use of an auxiliary index-based variable. Several numerical examples are presented showing favorable comparisons with MATCONT near saddle homoclinic points. The algorithm is also formulated with infinitesimal parameter increments resulting in ordinary differential equations, which gives some advantages like the ability to handle fold points of periodic solution branches upon suitable re-parametrization. Extensions to higher dimensions are outlined as well. With the new algorithm, we revisit the amplitude control system and continue the limit cycles much closer to the homoclinic point. We also provide some independent semi-analytical estimates of the homoclinic point, and mention an a typical property of the homoclinic orbit. In the second part of this thesis we analytically study the classical van der Pol oscillator, but with an added fractional damping term. We use the MMS near the Hopf bifurcation point. Systems with (1)fractional terms, such as the one studied here, have hitherto been largely treated numerically after suitable approximations of the fractional order operator in the frequency domain. Analytical progress has been restricted to systems with small fractional terms. Here, the fractional term is approximated by a recently pro-posed Galerkin-based discretization scheme resulting in a set of ODEs. These ODEs are then treated by the MMS, at parameter values close to the Hopf bifurcation. The resulting slow flow provides good approximations to the full numerical solutions. The system is also studied under weak resonant forcing. Quasiperiodicity, weak phase locking, and entrainment are observed. An interesting observation in this work is that although the Galerkin approximation nominally leaves several long time scales in the dynamics, useful MMS approximations of the fractional damping term are nevertheless obtained for relatively large deviations from the nominal bifurcation point. In the third part of this thesis, we study a well known tool vibration model in the large delay regime using the MMS. Systems with small delayed terms have been studied extensively as perturbations of harmonic oscillators. Systems with (1) delayed terms, but near Hopf points, have also been studied by the method of multiple scales. However, studies on systems with large delays are few in number. By “large” we mean here that the delay is much larger than the time scale of typical cutting tool oscillations. The MMS up to second order, recently developed for such large-delay systems, is applied. The second order analysis is shown to be more accurate than first order. Numerical integration of the MMS slow flow is much faster than for the original equation, yet shows excellent accuracy. A key point is that although certain parameters are treated as small(or, reciprocally, large), the analysis is not restricted to infinitesimal distances from the Hopf bifurcation. In the present analysis, infinite dimensional dynamics is retained in the slow flow, while the more usual center manifold reduction gives a planar phase space. Lower-dimensional dynamical features, such as Hopf bifurcations and families of periodic solutions, are also captured by the MMS. The strong sensitivity of the slow modulation dynamics to small changes in parameter values, peculiar to such systems with large delays, is seen clearly. In the last part of this thesis, we study the weakly nonlinear whirl of an asymmetric, overhung rotor near its gravity critical speed using a well known two-degree of freedom model. Gravity critical speeds of rotors have hitherto been studied using linear analysis, and ascribed to rotor stiffness asymmetry. Here we present a weakly nonlinear study of this phenomenon. Nonlinearities arise from finite displacements, and the rotor’s static lateral deflection under gravity is taken as small. Assuming small asymmetry and damping, slow flow equations for modulations of whirl amplitudes are developed using the MMS. Inertia asymmetry appears only at second order. More interestingly, even without stiffness asymmetry, the gravity-induced resonance survives through geometric nonlinearities. The gravity resonant forcing does not influence the resonant mode at leading order, unlike typical resonant oscillations. Nevertheless, the usual phenomena of resonances, namely saddle-node bifurcations, jump phenomena and hysteresis, are all observed. An unanticipated periodic solution branch is found. In the three dimensional space of two modal coefficients and a detuning parameter, the full set of periodic solutions is found to be an imperfect version of three mutually intersecting curves: a straight line, a parabola, and an ellipse. To summarize, the first and fourth problems, while involving routine MMS involve new applications with rich dynamics. The second problem demonstrated a semi-analytical approach via the MMS to study a fractional order system. Finally, the third problem studied a known application in a hitherto less-explored parameter regime through an atypical MMS procedure. In this way, a variety of problems that showcase the utility of the MMS have been studied in this thesis.

This thesis is divided into three parts. In the first part we study mixed norm estimates for Riesz transforms associated with various differential operators. First we prove the mixed norm estimates for the Riesz transforms associated with Dunkl harmonic oscillator by means of vector valued inequalities for sequences of operators defined in terms of Laguerre function expansions. In certain cases, the result can be deduced from the corresponding result for Hermite Riesz transforms, for which we give a simple and an independent proof. The mixed norm estimates for Riesz transforms associated with other operators, namely the sub-Laplacian on Heisenberg group, special Hermite operator on C^d and Laplace-Beltrami operator on the group SU(2) are obtained using their L^pestimates and by making use of a lemma of Herz and Riviere along with an idea of Rubio de Francia. Applying these results to functions expanded in terms of spherical harmonics, we deduce certain vector valued inequalities for sequences of operators defined in terms of radial parts of the corresponding operators. In the second part, we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian ∆_κ on weighted L^P-spaces. In the general case, for the chaotic behavior of the Dunkl-heat semigroup on weighted L^p-spaces, we only have partial results, but in the case of the heat semigroup generated by the standard Laplacian, a complete picture of the chaotic behavior is obtained on the spaces L^p ( R^d,〖 (φ_iρ (x ))〗^2 dx) where φ_iρ the Euclidean spherical function is. The behavior is very similar to the case of the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar. In the last part, we study mixed norm estimates for the Cesáro means associated with Dunkl-Hermite expansions on〖 R〗^d. These expansions arise when one considers the Dunkl-Hermite operator (or Dunkl harmonic oscillator)〖 H〗_κ:=-Δ_κ+|x|^2. It is shown that the desired mixed norm estimates are equivalent to vector-valued inequalities for a sequence of Cesáro means for Laguerre expansions with shifted parameter. In order to obtain the latter, we develop an argument to extend these operators for complex values of the parameters involved and apply a version of Three Lines Lemma.