The Schrödinger equation ... is considered. The solution of this equation is reduced to the problem of finding the eigenvectors of an infinite matrix. The infinite matrix is truncated to a finite matrix. The approximation due to the truncation is carefully studied. The band structure of the eigenvalues is shown. The eigenvectors of the multiwells potential are presented. The solutions of Schrödinger equation are calculated. The results are very sensitive to the value of the parameter y. Localized solutions, in the case that the energy is slightly greater than the maximum value of the potential, are presented. Wigner and Weyl functions, corresponding to the solutions of Schrödinger equation, are also studied. It is also shown that they are very sensitive to the value of the parameter y.

This thesis demonstrates how the geometric connection between the integrable Heisenberg spin equation, the nonlinear Schr??dinger equation and fluid flows with constant velocity magnitude along individual streamlines may be exploited. Specifically, we are able to construct explicitly the complete class of constant speed flows where the constant pressure surfaces constitute surfaces of revolution. This class is undoubtedly important as it contains many of the specific cases discussed earlier by other authors.

This thesis demonstrates how the geometric connection between the integrable Heisenberg spin equation, the nonlinear Schr??dinger equation and fluid flows with constant velocity magnitude along individual streamlines may be exploited. Specifically, we are able to construct explicitly the complete class of constant speed flows where the constant pressure surfaces constitute surfaces of revolution. This class is undoubtedly important as it contains many of the specific cases discussed earlier by other authors.

A prescription is given to obtain some exact results for certain external potentials �� (r) of the time-independent Gross-Pitaevskii and Schrodinger equations. The study motivation is the ability to program �� (r) experimentally in cold atom Bose-Einstein condensates. Rather than derive wavefunctions that are solutions for a given �� (r), we ask which �� (r) will have a given pdf (probability density function) �� (r). Several examples in 1 dimension (1D), 2 dimensions (2D), and 3 dimensions (3D) are presented for well-known pdfs in the position space. Exact potentials with zero, one and two walls are obtained and explained in detail. Apart from position space, the method is also applicable to obtain exact solutions for the Time-independent Schr¨odinger equation (TISE) and Gross-Pitaevskii equation (GPeq) for pdfs in momentum space. For this, we derived the potentials which are generated from the pdfs of the hydrogen atom in the real space as well as in the momentum space. However, the method was also extended for the time-dependent case. The prescription is also applicable to solve time-dependent pdfs. The aim is to find the ��(r, ��) which generates the pdf ��(r, ��). As a special case, we tested our method by studying the well known case for the Gaussian wave packet in 1D with zero potential ��(��, ��) = 0.

Made available in DSpace on 2014-12-17T14:13:36Z (GMT). No. of bitstreams: 1 EdjaneMAB.pdf: 131472 bytes, checksum: fa624d7510ee901925a2df6063e51778 (MD5) Previous issue date: 2008-07-30 / Fucan is a term used to denominate a family of sulfated L-fucose-rich polysaccharides. The brown alga Spatoglossum schr?ederi (Dictyotaceae) has three heterofucans namely fucan A, B and C. The 21 kDa fucan A is composed of a core of β (1-3) glucuronic acid-containing oligosaccharide of 4.5 kDa with branches at C4 of fucose chains α (1-3) linked. The fucose is mostly substituted at C4 with a sulfate group and at C2 with chains of β (1-4) xylose. This fucan has neither anticoagulant (from from 0.1 to 100?g) nor hemorrhagic activities (from 50 to 800 ?g/mL). The antithrombotic test in vivo showed the fucan A has no activity in any of the concentrations (from 0.2 to 20?g/g/day) tested 1h after polysaccharide administration. However, when fucan A was injected endovenously 24h before the ligature of the venae cavae, we observed a dose-dependent effect, reaching saturation at around 20g/g of rat weight. In addition, this effect is also time-dependent, reaching saturation around 16h after fucan administration. In addition, regardless of administration pathway, fucan A displayed antithrombotic action. The exception was the oral pathway. Of particular importance was the finding that fucan A stimulates the synthesis of an antithrombotic heparan sulfate from endothelial cells like heparin. The hypothesis has been raised that in vivo antithrombotic activity of fucan A is related to the increased production this heparan. Taken together with the fact that the compound is practically devoid of anticoagulant and hemorrhagic activity suggests that it may be an ideal antithrombotic agent in vivo / Fucanas ? um termo utilizado para denominar a fam?lia de polissacar?deos sulfatados que apresentam em sua constitui??o α-L-fucose sulfatada. A alga marrom Spatoglossum schr?ederi da fam?lia Dictyotaceae, apresenta tr?s heterofucanas denominadas de fucana A, B, e C. A fucana A (21kDa) ? composta por um n?cleo central formado por ?cido glucur?nico, β 1→3 ligado, substitu?dos em C4 por L-fucoses α (1→3) ligados. A fucose ainda ? substitu?da no C4 por grupos sulfatos e no C2 por cadeias de β (1→4) xilose. Esta fucana n?o apresentou atividade anticoagulante entre 0,1 e 100?g/mL e nenhuma atividade hemorr?gica entre 50 e 800?g /mL. Os testes antitromb?tico in vivo mostraram que a fucana A n?o apresentou atividade nas concentra??es (0,2 a 20?g/g) testadas 1 hora ap?s a administra??o do polissacar?deo. No entanto, quando a fucana A foi administrada endovenosamente, 24h antes da ligadura da veia cava, observou-se um efeito dose-dependente, alcan?ando a sua satura??o em torno de 20?g/g de massa do animal. Al?m disso, o efeito se mostrou tempo-dependente, atingindo satura??o em 16h ap?s a administra??o da fucana. Em todas as vias em que foi administratada (SC, IM, IP, e IV), a fucana A demonstrou a??o antitromb?tica. Exce??o apenas para a via oral. A import?ncia dos resultados obtidos foi o fato da fucana A estimular a s?ntese de um heparam sulfato com caracter?sticas antitromb?tica semelhante ? heparina pelas c?lulas endoteliais. Esta descoberta levou a hip?tese de que a atividade antitromb?tica da fucana A est? relacionada com o aumento na produ??o deste heparam sulfato. Devido as suas caracter?sticas e por ser um composto praticamente desprovido de atividade anticoagulante e hemorr?gico sugere-se que a fucana A possa vir a ser um agente antitromb?tico in vivo . A realiza??o deste estudo teve car?ter multidisciplinar, envolvendo pesquisadores da Bioqu?mica, Morfologia, Hematologia, Bot?nica e Veterin?ria. Este aspecto preencheu os requisitos da multidisciplinaridade do Programa de P?s-Gradua??o em Ci?ncias da Sa?de

In this thesis, I study the group theoretic structure of the Schrodinger equations of simple systems by making use of a new systematic method. Group theoretic analysis of Schrodinger equations have been made previously by numerous physicists. The groups found may be classified as: a) geometrical groups; b) dynamical degeneracy groups; c) dynamical groups The geometrical group arises simply from the spatial symmetry of the system. Although the geometrical groups are very useful, they are not very interesting from the physical viewpoint. On the other hand, the study of the dynamical degeneracy groups and the dynamical group is very attractive because it reflects the dynamic of the system. Extensive studies have previously been made by other authors on systems which exhibit nontrivial degeneracy (accidental degeneracy). It turns out that all the states which belong to the same energy level provide a basis for a unitary irreducible representation of some compact group, and the group itself is generated by a set of constants of the motion. These groups are called “dynamical degeneracy groups”. Detailed discussion on degeneracy groups will be found in the paper by McIntosh alluded to above.

Multisymplectic integrators like Preissman and six-point schemes and a semi-explicit symplectic method are applied to the coupled nonlinear Schr&ouml / dinger equations (CNLSE). Energy, momentum and additional conserved quantities are preserved by the multisymplectic integrators, which are shown using modified equations. The multisymplectic schemes are backward stable and non-dissipative. A semi-explicit method which is symplectic in the space variable and based on linear-nonlinear, even-odd splitting in time is derived. These methods are applied to the CNLSE with plane wave and soliton solutions for various combinations of the parameters of the equation. The numerical results confirm the excellent long time behavior of the conserved quantities and preservation of the shape of the soliton solutions in space and time.

In this thesis, the main perturbation problems encountered in quantum mechanics have been studied.Since the special functions and orthogonal polynomials appear very extensively in such problems, we emphasize on those topics as well. In this context, the classical quantum mechanical anharmonic oscillators described mathematically by the one-dimensional Schr&uml / odinger equation have been treated perturbatively in both finite and infinite intervals, corresponding to confined and non-confined systems, respectively.

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

In this thesis, a general pseudospectral formulation for a class of Sturm-Liouville eigenvalue problems is consructed. It is shown that almost all, regular or singular, Sturm-Liouville eigenvalue problems in the Schr&ouml / dinger form may be transformed into a more tractable form. This tractable form will be called here a weighted equation of hypergeometric type with a perturbation (WEHTP) since the non-weighted and unperturbed part of it is known as the equation of hypergeometric type (EHT). It is well known that the EHT has polynomial solutions which form a basis for the Hilbert space of square integrable functions. Pseudospectral methods based on this natural expansion basis are constructed to approximate the eigenvalues of WEHTP, and hence the energy eigenvalues of the Schr&ouml / dinger equation. Exemplary computations are performed to support the convergence numerically.