RESUMO Apresenta-se uma análise crítica de uma técnica numérica que tem sido usada na resolução de equações diferenciais de ordem fracionária com derivadas de Caputo. Trata-se do método multi-passos com transformada diferencial generalizada. Verifica-se que a versão do método disponível na literatura produz soluções erradas a partir do segundo passo, isto é mostrado em aplicações aos modelos de Malthus e de Riccati. O problema é explicado em termos da não localidade da derivada de Caputo e das propriedades da transformada diferencial generalizada.

ABSTRACT This paper presents a critical analysis of a numerical technique that has been used in the resolution of fractional differential equations with Caputo derivative. It is the multi-step method with generalized differential transform. It is found that the version of the method available in the literature produces erroneous solutions from the second step, this is shown in applications to the Malthus and Riccati models. The procedure error is explained in terms of the nonlocality of the Caputo derivative and the properties of the generalized differential transform.

RESUMO Este trabalho aborda o problema da diagonalização conjunta aproximada de uma coleção de matrizes reais e simétricas. A otimização é realizada com a restrição de que a matriz de transformação de semelhança seja ortogonal. As soluções são apresentadas de forma analítica para matrizes de ordem 2. São introduzidos os conceitos de vetor anti-diagonalizante, amplitude de uma matriz, que é expressa em termos dos autovalores, e matrizes parcialmente complementares. Isto permite fazer uma interpretação geométrica da diagonalização conjunta aproximada, em termos dos autovetores e dos vetores anti-diagonalizantes das matrizes. Esta contribuição deve auxiliar na melhoria de procedimentos numéricos e computacionais envolvendo matrizes de ordem maior que 2.

ABSTRACT The problem of joint approximate diagonalization of symmetric real matrices is addressed. It is reduced to an optimization problem with the restriction that the matrix of the similarity transformation is orthogonal. Analytical solutions are derived for the case of matrices of order 2. The concepts of off-diagonalizing vectors, matrix amplitude, which is given in terms of the eigenvalues, and partially complementary matrices are introduced. This leads to a geometrical interpretation of the joint approximate diagonalization in terms of eigenvectors and off-diagonalizing vectors of the matrices. This should be helpful to deal with numerical and computational procedures involving high-order matrices.

Stationary states of an electron in thin GaAs elliptical quantum rings are calculated within the effective-mass approximation. The width of the ring varies smoothly along the centerline, which is an ellipse. The solutions of the Schrödinger equation with Dirichlet boundary conditions are approximated by a product of longitudinal and transversal wave functions. The ground-state probability density shows peaks: (i) where the curvature is larger in a constant-with ring, and (ii) in thicker parts of a circular ring. For rings of typical dimensions, it is shown that the effects of a varying width may be stronger than those of the varying curvature. Also, a width profile which compensates the main localization effects of the varying curvature is obtained.

Coherent properties and Rabi oscillations in two-level donor systems, under terahertz excitation, are theoretically investigated. Here we are concerned with donor states in bulk GaAs and GaAs - (Ga, Al)As quantum dots. We study confinement effects, in the presence of an applied magnetic field, on the electronic and on-center donor states in GaAs - (Ga, Al)As dots, as compared to the situation in bulk GaAs, and estimate some of the associated decay rate parameters. Using the optical Bloch equations with damping, we study the time evolution of the 1s and 2p+ states in the presence of an applied magnetic field and of a terahertz laser. We also discuss the role played by the distinct dephasing rates on the photocurrent and calculate the electric dipole transition moment. Results indicate that the Rabi oscillations are more robust as the total dephasing rate diminishes, corresponding to a favorable coherence time.