Solutions Of Eigenvalue Equation Research Articles

Método aproximado para determinar la potencia óptica en una linterna fotónica

Astronomy has benefited significantly from the development of photonic technology. However, the use of single-mode optical fibers in this area is not entirely efficient, this is mainly since its core, of the order of 8 microns in diameter, does not allow the capture of large amounts of light. In addition, in certain astronomical studies, it is required to analyze the multimodal incoherent light coming from the stars, particularly in the spectral range of the near infrared. One solution to these demands is the use of photonic lanterns, which are devices that function as an interface between a set of single-mode fibers and a multimode fiber, and in whose transition very low optical losses are obtained. However, there is not as far as we know, a mathematical method for the analysis of the behavior of the output power in photonic lanterns. Therefore, in this work we propose a numerical approximation method to determine the optical output power of single mode optical fibers in a photonic lantern, using the solution of eigenvalue equations, as well as the spatial capture of a part of the distribution of optical power, through of the end of a monomode tapered optical fiber.

Reduced-Order Modeling for and Vibration Characteristics Analysis of a Hard-Coated Mistuned Blisk

This paper develops a damping strategy for the vibration reduction of a mistuned bladed disk (blisk) by depositing hard coating on its blades, and systematically investigates the vibration characteristics of the hard-coated mistuned (HCM) blisk. By using an improved fixed-interface component modal synthesis method (fixed-ICMSM), a reduced-order model (ROM) of the HCM blisk is established. Then, based on the proposed ROM, solutions of eigenvalue equations are carried out to obtain the natural frequencies and mode shapes. Further, modal loss factors and a damping matrix of the HCM blisk are achieved by taking advantage of the modal strain energy method and the proportional damping model, respectively. Moreover, the frequency response function of the HCM blisk, which can exhibit dynamic behaviors, was deduced. Finally, a mistuned blisk with a deposited NiCoCrAlY + YSZ hard coating on both sides of the blades is chosen as a study case to conduct a finite element analysis, and the results are compared with those obtained from the experimental test in terms of natural frequencies and mode shapes. The variation of the natural frequencies, the modal loss factors, and the frequency response function generated by the NiCoCrAlY + YSZ hard coating are studied, and the influence of coating area on damping capacity is further discussed.

The eigenvalue equations of equatorial waves on a sphere

Zonally propagating wave solutions of the linearized shallow water equations (LSWE) in a zonal channel on the rotating spherical earth are constructed from numerical solutions of eigenvalue equations that yield the meridional variation of the waves’ amplitudes and the phase speeds of these waves. An approximate Schr¨odinger equation, whose potential depends on one parameter only, is derived, and this equation yields analytic expressions for the dispersion relations and for the meridional structure of the waves’ amplitudes in two asymptotic cases. These analytic solutions validate the accuracy of the numerical solutions of the exact eigenvalue equation. Our results show the existence of Kelvin, Poincar´e and Rossby waves that are harmonic for large radius of deformation. For small radius of deformation, the latter two waves vary as Hermite functions. In addition, our results show that the mixed mode of the planar theory (a meridional wavenumber zero mode that behaves as a Rossby wave for large zonal wavenumbers and as a Poincar´e wave for small ones) does not exist on a sphere; instead, the first Rossby mode and the first westward propagating Poincar´e mode are separated by the anti-Kelvin mode for all values of the zonal wavenumber.

Numerical electronic structure calculations for atoms. I. Generalized variable transformation and nonrelativistic calculations

The radial functions Pi in the nonrelativistic description of the electronic structure of atoms are solutions of eigenvalue equations. These equations are treated as two-point boundary value problems for bound one-electron states and can be solved to high accuracy with finite difference methods. For these numerical techniques the transformation to a suitable new radial variable is essential. The effects and consequences following from an arbitrary suitable variable transformation are studied in the most general way. Important results following from this general analysis are the extension of the range of application of the standard Numerov scheme and the outline for an algorithm of increased overall efficiency and reliability. Especially, the explicit use of transformed solution functions is shown to be unnecessary. Radial functions can be determined for arbitrary effective potentials resulting from the underlying theoretical description. It is demonstrated that all numerical results can be calculated to within a consistent numerical truncation error of order h4 in the grid point spacing. The approach can be extended to handle unbound one-electron states and is applicable in other fields, where differential equations of similar character occur, e.g., in the description of the vibration of a diatomic molecule. A similar analysis for relativistic electronic structure calculations for atoms will be given in Part II. © 1997 John Wiley & Sons, Inc.

Numerical electronic structure calculations for atoms. I. Generalized variable transformation and nonrelativistic calculations

The radial functions Pi in the nonrelativistic description of the electronic structure of atoms are solutions of eigenvalue equations. These equations are treated as two-point boundary value problems for bound one-electron states and can be solved to high accuracy with finite difference methods. For these numerical techniques the transformation to a suitable new radial variable is essential. The effects and consequences following from an arbitrary suitable variable transformation are studied in the most general way. Important results following from this general analysis are the extension of the range of application of the standard Numerov scheme and the outline for an algorithm of increased overall efficiency and reliability. Especially, the explicit use of transformed solution functions is shown to be unnecessary. Radial functions can be determined for arbitrary effective potentials resulting from the underlying theoretical description. It is demonstrated that all numerical results can be calculated to within a consistent numerical truncation error of order h4 in the grid point spacing. The approach can be extended to handle unbound one-electron states and is applicable in other fields, where differential equations of similar character occur, e.g., in the description of the vibration of a diatomic molecule. A similar analysis for relativistic electronic structure calculations for atoms will be given in Part II. © 1997 John Wiley & Sons, Inc.

Efficient solution of eigenvalue equations of optical waveguiding structures

A general numerical method is presented that is capable of extracting all the zeros of a complex equation. This method allows accurate determination of the complex propagation constants of general multilayer optical and microwave planar waveguides, the computation of energy states and their lifetimes of semiconductor heterostructures, and the roots of complex transcendental equations from other scientific disciplines. The method differs from previous approaches in that the time-consuming differentiation of the complex function is not required. Furthermore, the present method is not affected by the presence of complex poles and so can be used for the solution of meromorphic transcendental equations. In practice, the method is found to be much faster than other rigorous approaches. >

Solving the Schr�dinger Equation with a Hand-Held Calculator

A rather simple method of numerical integration, the Euler method, is applied to the one-dimensional Schrodinger equation for a paticle bound to a finite square well, a case which is exactly solvable by analytical methods. Insight is gained into the dominant role played by the boundary conditions in the solution of eigenvalue equations. Comparison is made with the exact soution of the algebraic methods, which leads to the discussion of improving the method, or switching to more sophisticated procedures. Though the results presented came from calculations performed on a nonprogrammmable calculator, programmable ones could be used. The same treatmnent can be applied to more complex potentials, for example two-and three-dimensional ones. The beauty of the method is that the solution seems to come from the calculating fingers rather than from an abstract equation. Emphasis is on the physics rather than on the calculator.

Reduction of Group Representations. V

This paper shows how to use information which can be obtained by use of the Todd Coxeter algorithm and related techniques, to reduce a group representation. The method reduces a representation of G given a reduction for a subgroup H of finite index in G. If the character table for G is given, only rational algebraic processes and square roots are used. If the character table for G is not known it can be found by solution of eigenvalue equations, using the class sums, which are obtained in the course of the calculation without scanning the whole group.