PT-symmetric Hamiltonians proposed by Bender and Boettcher can have real energy spectra. As an extension of the Hermitian Hamiltonian, PT-symmetric systems have attracted a great interest in recent years. Understanding the underlying mathematical structure of these theories sheds insight on outstanding problems of physics. These problems include the nature of Higgs particles, the properties of dark matter, the matter-antimatter asymmetry in the universe, and neutrino oscillations. Furthermore, PT-phase transition has been observed in lasers, optical waveguides, microwave cavities, superconducting wires and circuits. The objective of this thesis is to extend the iterative method of solving Schrodinger equation used for an harmonic oscillator systems to Hamiltonians with PT-symmetric potentials. An important aspect of this approach is the high accuracy of eigenvalues and the fast convergence. Our method is a combination of Hill determinant method [8] and the power series expansion. eigenvalues and the fast convergence. One can transform the Schrodinger equation into a secular equation by using a trial wave function. A recursion structure can be obtained using the secular equation, which leads to accurate eigenvalues. Energy values approach to exact ones when the number of iterations is increased. We obtained eigenvalues for a set of PT-symmetric Hamiltonians.

Orientador: José Francisco Gomes / Co-orientador: Abraham Hirsz Zimerman / Banca: Antônio Lima Santos / Banca: Clisthenis Ponce Constantinidis / Resumo: A teoria de defeitos integráveis em teoria de campos em 1+1 dimensões, foi introduzida pela escola de York [16, 17, 22], utilizando transformações de Bäcklund para descrever o defeito. Nesta dissertação estudamos o modelo de Schrödinger não linear na presença de um defeito integrável. Estudamos tanto o modelo discreto [29] como o modelo contínuo dentro dos formalismos lagrangiano [23] e da matrizr [7]. Construímos também o formalismo hamiltoniano para o modelo de Schrödinger não linear na presença de um defeito integrável. Discutimos e relacionamos os formalismos lagrangiano, hamiltoniano e da matriz r / Abstract: The theory of integrable defects in 1+1 field theory, was introduced by the school of York [16, 17, 22], employing B¨acklund transformation in order to describe the defect. In this dissertation we have studied the nonlinear Schrödinger model in the presence of an integrable defect. We study both, the discrete model [29] as the continuous model within the lagrangian [23] andr matrix [7] formalisms. Also we built the hamiltonian formalism for nonlinear Schrödinger model in the presence of an integrable defect. We discuss and relate the lagrangian, hamiltonian and r matrix formalisms / Mestre

In this thesis, we will study non-linear dispersive equations. The primary focus will be on the construction of the positive-time wave operator for such equations. The positive-time wave operator problem arises in the study of the asymptotics of a partial differential equation. It is a map from a space of initial data X into itself, and is loosely defined as follows: Suppose that for a solution Ψlin to the dispersive equation with no non-linearity and initial data Ψ+ there exists a unique solution Ψ to the non-linear equation with initial data ΨO such that Ψ behaves as Ψlin as t→ ∞. Then the wave operator is the map W + that takes Ψ+/sub; to Ψ0. By its definition, W+ is injective. An important additional question is whether or not the map is also surjective. If so, then every non-linear solution emanating from X behaves, in some sense, linearly as it evolves (this is known as asymptotic completeness). Thus, there is some justification for treating these solutions as their much simpler linear counterparts. The main results presented in this thesis revolve around the construction of the wave operator(s) at critical non-linearities. We will study the #8220; semi-relativistic ” Schrëdinger equation as well as the Klein-Gordon-Schrëdinger system on R2. In both cases, we will impose fairly general quadratic non-linearities for which conservation laws cannot be relied upon. These non-linearities fall below the scaling required to employ such tools as the Strichartz estimates. We instead adapt the "first iteration method" of Jang, Li, and Zhang to our setting which depends crucially on the critical decay of the non-linear interaction of the linear evolution. To see the critical decay in our problem, careful analysis is needed to treat the regime where one has spatial and/or time resonance.

The eigenvalue gaps and eigenvalue ratios of the Sturm-Liouville systems have been studied in many papers. Recently, Lavine proved an optimal lower estimate of first eigenvalue gaps for Schrodinger operators with convex potentials. His method uses a variational approach with detailed analysis on different integrals. In 1999, (M.J.) Huang adopted his method to study eigenvalue ratios of vibrating strings. He proved an optimal lower estimate of first eigenvalue ratios with nonnegative densities. In this thesis, we want to generalize the above optimal estimate. The work of Ashbaugh and Benguria helps in attaining our objective. They introduced an approach involving a modified Prufer substitution and a comparison theorem to study the upper bounds of Dirichlet eigenvalue ratios for Schrodinger operators with nonnegative potentials. It is interesting to see that the counterpart of their result is also valid. By Liouville substitution and an approximation theorem, the vibrating strings with concave and positive densities can be transformed to a Schrodinger operator with nonpositive potentials. Thus we have the generalization of Huang's result.

published_or_final_version / Physics / Master / Master of Philosophy

No description available.

In the first part of this thesis, we demonstrate theory and computations for finite-energy line defect solutions in an improvement of Ericksen-Leslie liquid crystal theory. Planar director fields are considered in two and three space dimensions, and we demonstrate straight as well as loop disclination solutions. The possibility of static balance of forces in the presence of a disclination and in the absence of ow and body forces is discussed. The work exploits an implicit conceptual connection between the Weingarten-Volterra characterization of possible jumps in certain potential fields and the Stokes-Helmholtz resolution of vector fields. The theoretical basis of our work is compared and contrasted with the theory of Volterra disclinations in elasticity. Physical reasoning precluding a gauge-invariant structure for the model is also presented. In part II of the thesis, the time-harmonic Schrodinger equation with periodic potential is considered. We derive the asymptotic form of the scattering wave function in the periodic space and investigate the possibility of its application as a DtN non-reflecting boundary condition. Moreover, we study the perfectly matched layer method for this problem and show that it is a reliable method, which converges rapidly to the exact solution, as the thickness of the absorbing layer increases. Moreover, we use the tight-binding method to numerically solve the Schrodinger equation for Graphene sheets, symmetry-adapted Carbon nanotubes and DNA molecules to demonstrate their electronic behavior in the presence of local defects. The results for Y-junction Carbon nanotubes depict very interesting properties and confirms the predictions for their application as new transistors.

Internal gravity waves grow in amplitude as they propagate upwards in a non-Boussinesq fluid and weakly nonlinear effects develop due to interactions with an induced horizontal mean flow. In this work, a new derivation for this wave-induced mean flow is presented and nonlinear Schrodinger equations are derived describing the weakly nonlinear evolution of these waves in an anelastic gas and non-Boussinesq liquid. The results of these equations are compared with fully nonlinear numerical simulations. It is found that interactions with the wave-induced mean flow are the dominant mechanism for wave evolution. This causes modulational stability for hydrostatic waves, resulting in propagation above the overturning level predicted by linear theory for a non-Boussinesq liquid. Due to high-order dispersion terms in the Schrodinger equation for an anelastic gas, hydrostatic waves become unstable and break at lower levels. Non-hydrostatic waves are modulationally unstable, overturning at lower levels than predicted by linear theory.

Orientador: Bin Kang Cheng / Dissertação (mestrado) - Universidade Federal do Paraná / Resumo: Neste trabalho de tese estudamos inicialmente o formalismo quântico do potencial com n funções S de Dirac. Em seguida, obtemos os resultados quânticos resolvendo a equação de Schrödinger independente do tempo e usando o método de integral de caminhos. Estudamos dois casos particulares do potencial: (a) potencial não confinado com ri — 2 e (b) potencial confinado com n — 41. Pela solução da equação de Schrödinger independente do tempo podemos determinar completamente a função de onda e os autovalores da energia. Com a integral de caminho obtemos as funções de Green exatas. Para o potencial confinado com n = 4 temos também o traço da função de Green, da qual calculamos os níveis quânticos que estão em total acordo com os resultados obtidos da solução da equação de Schrödinger. / Abstract: In this thesis we first study the quantum formalism of the potential comprising of n Dirac S functions. We present the quantum results both by solving the time independent Schrödinger’s equation and by using the path integral method. We investigate two particular cases of the potential: (a) unbound potential with n = 2 and (b) bound potential with n = 42. From the Schrödinger equation we in fact determine completely the wave function and their energy levels. Applying the path integral method we obtain the exact Green functions and for the bound potential with n = 4 we also obtain the trace of the Green function, from which we evaluate the quantized energies which are in fully agreement with those obtained by solving the Schrödinger’s equation.

Made available in DSpace on 2014-06-11T19:27:08Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-02-28Bitstream added on 2014-06-13T19:14:20Z : No. of bitstreams: 1 bazao_vr_me_prud.pdf: 2551094 bytes, checksum: 6407adf5649c80273f9cd3097f312d5f (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Neste trabalho fizemos um levantamento das diferentes versões discretas e contínuas dos argumentos de Gordon, utilizados no estudo espectral de operadores de Schrödinger unidimensionais. Estudamos como aproximações periódicas do potencial (caso contínuo) e ocorrências de estruturas repetitivas do potencial (caso discreto) permitem excluir o espectro pontual de tais operadores. No caso discreto, as aplicações dos argumentos de Gordon fornecem resultados genéricos, q.t.p. (quase toda parte) e uniformes sobre a ausência de espectro pontual para modelos de Schrödinger com potenciais gerados por substituições primitivas e rotações na circunferência. Parte dos resultados obtidos na demonstração desses argumentos podem ser usados para mostrar que o espectro dos operadores tem medida de Lebesgue zero. Consequentemente, com a ocorrência simultânea das propriedades ausência de espectro pontual e espectro com medida zero , obtemos operadores de Schrödinger com espectro puramente singular contínuo. No caso contí- nuo, as aplicações incluem operadores de Schrödinger gerados por potenciais de Gordon com frequências de Liouville, funções Hölder contínuas, funções escada e funções com singularidades locais / In this work review di erent versions of discrete and continuous Gordon's arguments, used in the spectral study of one-dimensional Schrödinger operators. We study periodic approximations of the potential (continuous case) and occurrences of repetitive structures of the potential (discrete case) that allow us to exclude the point spectrum of such operators. In the discrete case, the applications of Gordon's arguments supply generic results, almost sure and uniform on the absence of point spectrum for Schrödinger models with potentials generated by primitive substitutions and circle maps. Part of the results obtained in the demonstration of these arguments can be used to show that the spectrum of the operators has zero Lebesgue measure. Consequently, with the properties absence of point spectrum and spectrum with zero measure , we obtain Schrödinger operators with purely singular continuous spectrum. In the continuous case, the applications include Schrödinger operators generated by Gordon potentials with Liouville frequencies, Hölder continuous functions, step functions and functions with power-type singularities