The pseudospectral method is a family of numerical methods for the solution of differential equations based on the expansion of basis functions defined on a set of grid points. In this thesis, the relationship between the distribution of grid points and the accuracy and convergence of the solution is emphasized. The polynomial and sinc pseudospectral methods are extensively studied along with many applications to quantum and statistical mechanics involving the Fokker-Planck and Schroedinger equations. The grid points used in the polynomial methods coincide with the points of quadrature, which are defined by a set of polynomials orthogonal with respect to a weight function. The choice of the weight function plays an important role in the convergence of the solution. It is observed that rapid convergence is usually achieved when the weight function is chosen to be the square of the ground-state eigenfunction of the problem. The sinc method usually provides a slow convergence as the grid points are uniformly distributed regardless of the behaviour of the solution. For both polynomial and sinc methods, the convergence rate can be improved by redistributing the grid points to more appropriate positions through a transformation of coordinates. The transformation method discussed in this thesis preserves the orthogonality of the basis functions and provides simple expressions for the construction of discretized matrix operators. The convergence rate can be improved by several times in the evaluation of loosely bound eigenstates with an exponential or hyperbolic sine transformation. The transformation can be defined explicitly or implicitly. An explicit transformation is based on a predefined mapping function, while an implicit transformation is constructed by an appropriate set of grid points determined by the behaviour of the solution. The methodologies of these transformations are discussed with some applications to 1D and 2D problems. The implicit transformation is also used as a moving mesh method for the time-dependent Smoluchowski equation when a function with localized behaviour is used as the initial condition.

The pseudospectral method is a family of numerical methods for the solution of differential equations based on the expansion of basis functions defined on a set of grid points. In this thesis, the relationship between the distribution of grid points and the accuracy and convergence of the solution is emphasized. The polynomial and sinc pseudospectral methods are extensively studied along with many applications to quantum and statistical mechanics involving the Fokker-Planck and Schroedinger equations. The grid points used in the polynomial methods coincide with the points of quadrature, which are defined by a set of polynomials orthogonal with respect to a weight function. The choice of the weight function plays an important role in the convergence of the solution. It is observed that rapid convergence is usually achieved when the weight function is chosen to be the square of the ground-state eigenfunction of the problem. The sinc method usually provides a slow convergence as the grid points are uniformly distributed regardless of the behaviour of the solution. For both polynomial and sinc methods, the convergence rate can be improved by redistributing the grid points to more appropriate positions through a transformation of coordinates. The transformation method discussed in this thesis preserves the orthogonality of the basis functions and provides simple expressions for the construction of discretized matrix operators. The convergence rate can be improved by several times in the evaluation of loosely bound eigenstates with an exponential or hyperbolic sine transformation. The transformation can be defined explicitly or implicitly. An explicit transformation is based on a predefined mapping function, while an implicit transformation is constructed by an appropriate set of grid points determined by the behaviour of the solution. The methodologies of these transformations are discussed with some applications to 1D and 2D problems. The implicit transformation is also used as a moving mesh method for the time-dependent Smoluchowski equation when a function with localized behaviour is used as the initial condition.

Made available in DSpace on 2014-06-11T19:32:09Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-02-04Bitstream added on 2014-06-13T21:03:27Z : No. of bitstreams: 1 santos_vgcs_dr_guara.pdf: 603356 bytes, checksum: 48d0890069648043a713c383f62ba614 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / 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. / 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.

The pseudospectral method is a family of numerical methods for the solution of differential equations based on the expansion of basis functions defined on a set of grid points. In this thesis, the relationship between the distribution of grid points and the accuracy and convergence of the solution is emphasized. The polynomial and sinc pseudospectral methods are extensively studied along with many applications to quantum and statistical mechanics involving the Fokker-Planck and Schroedinger equations. The grid points used in the polynomial methods coincide with the points of quadrature, which are defined by a set of polynomials orthogonal with respect to a weight function. The choice of the weight function plays an important role in the convergence of the solution. It is observed that rapid convergence is usually achieved when the weight function is chosen to be the square of the ground-state eigenfunction of the problem. The sinc method usually provides a slow convergence as the grid points are uniformly distributed regardless of the behaviour of the solution. For both polynomial and sinc methods, the convergence rate can be improved by redistributing the grid points to more appropriate positions through a transformation of coordinates. The transformation method discussed in this thesis preserves the orthogonality of the basis functions and provides simple expressions for the construction of discretized matrix operators. The convergence rate can be improved by several times in the evaluation of loosely bound eigenstates with an exponential or hyperbolic sine transformation. The transformation can be defined explicitly or implicitly. An explicit transformation is based on a predefined mapping function, while an implicit transformation is constructed by an appropriate set of grid points determined by the behaviour of the solution. The methodologies of these transformations are discussed with some applications to 1D and 2D problems. The implicit transformation is also used as a moving mesh method for the time-dependent Smoluchowski equation when a function with localized behaviour is used as the initial condition. / Science, Faculty of / Mathematics, Department of / Graduate

In this thesis, we establish an analog of the Paley-Wiener Theorem for the ZS-AKNS scattering transform on a set of real potentials. We also demonstrate one application of our techniques to the study of an inverse spectral problem for a half-line Miura potential Schroedinger equation.

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This thesis presents a development of classical canonical formalism and the usual transition schema to quantum dynamics. The question of transition from relativistic mechanics to relativistic quantum dynamics is answered by developing a homogeneous formalism which is relativistically invariant. Using this formalism the Klein-Gordon equation is derived as the relativistic analog of the Schroedinger equation. Using this formalism further, a method of generating other relativistic equations (with spin) is presented.

This thesis is concerned with the Schrödinger representation of quantum field theory. We describe techniques for solving the Schrödinger equation which supplement the standard techniques of field theory. Our aim is to develop these to the point where they can readily be used to address problems of current interest. To this end, we study realistic models such as gauge theories coupled to dynamical fermions. For maximal generality we consider particles of all physical spins, in various dimensions, and eventually, curved spacetimes. We begin by considering Gaussian fields, and proceed to a detailed study of the Schwinger model, which is, amongst other things, a useful model for (3+1) dimensional gauge theory. One of the most important developments of recent years is a conjecture by Mal-dacena which relates supergravity and string/M-theory on anti-de-Sitter spacetimes to conformal field theories on their boundaries. This correspondence has a natural interpretation in the Schrödinger representation, so we solve the Schrödinger equation for fields of arbitrary spin in anti-de-Sitter spacetimes, and use this to investigate the conjectured correspondence. Our main result is to calculate the Weyl anomalies arising from supergravity fields, which, summed over the supermultiplets of type JIB supergravity compactified on AdS(_s) x S(^5) correctly matches the anomaly calculated in the conjecturally dual N = 4 SU{N) super-Yang-Mills theory. This is one of the few existing pieces of evidence for Maldacena's conjecture beyond leading order in TV.

A new intermolecular potential energy surface, rovibrational transition frequencies, and line strengths are computed for OCS-OCS and CO2-CS2. The potentials were made by fitting energies obtained from explicitly correlated coupled-cluster calculations and fit using an interpolating moving least squares method. Rovibrational transition frequencies are also calculated for four isotopologues of the N2O dimer using a previously presented potential energy surface. The rovibrational Schroedinger equation for all three dimers is solved with a symmetry-adapted Lanczos algorithm and an uncoupled product basis set. All four intermolecular coordinates are included in the calculation. On the OCS-OCS potential energy surface, a previously unknown, cross-shaped isomer is found along with polar and non-polar isomers. For CO2-CS2, the previously found cross-shaped minima is found along with a slipped-parallel configuration. The associated wavefunctions and energy levels for each of these isomers is presented. To identify states that have a permanent dipole, both calculations of line strengths and vibrational parent analysis is used. For non polar states of, OCS-OCS, and N2O-N2O isotopologues, and all CO2-CO2 states, only vibrational parent analysis was used. Calculated rotational constants differ from their experimental counterparts by less than 0.001 wavenumbers for OCS-OCS and CO2-CS2, and less than 0.002 wavenumbers for any N2O-N2O isotopologue. / Thesis (Master, Physics, Engineering Physics and Astronomy) -- Queen's University, 2012-08-23 13:19:45.294