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Francisco M. Fernandez

Publications and source records attributed to Francisco M. Fernandez.

At least 19 recordsLinked to original sources
arXiv2015

Weakly bound states in heterogeneous waveguides

We study the spectrum of the Helmholtz equation in a two-dimensional infinite waveguide, containing a weak heterogeneity localized at an internal point, and obeying Dirichlet boundary conditions at its border. We prove that, when the heterogeneity corresponds to a locally denser material, the lowest eigenvalue of the spectrum falls below the continuum threshold and a bound state appears, localized at the heterogeneity. We devise a rigorous perturbation scheme and derive the exact expression for the energy to third order in the heterogeneity.

math-ph↗
arXiv2015

High order eigenvalues for the Helmholtz equation in complicated non-tensor domains through Richardson Extrapolation of second order finite differences

We apply second order finite difference to calculate the lowest eigenvalues of the Helmholtz equation, for complicated non-tensor domains in the plane, using different grids which sample exactly the border of the domain. We show that the results obtained applying Richardson and Padé-Richardson extrapolation to a set of finite difference eigenvalues corresponding to different grids allows to obtain extremely precise values. When possible we have assessed the precision of our extrapolations comparing them with the highly precise results obtained using the method of particular solutions. Our empirical findings suggest an asymptotic nature of the FD series. In all the cases studied, we are able to report numerical results which are more precise than those available in the literature.

physics.comp-ph↗
arXiv2010

The confined hydrogen atom with a moving nucleus

We study the hydrogen atom confined to a spherical box with impenetrable walls but, unlike earlier pedagogical articles on the subject, we assume that the nucleus also moves. We obtain the ground-state energy approximately by means of first--order perturbation theory and by a more accurate variational approach. We show that it is greater than the one for the case in which the nucleus is clamped at the center of the box. Present approach resembles the well-known treatment of the helium atom with clamped nucleus.

quant-ph↗
arXiv2009

Perturbation approaches and Taylor series

We comment on the new trend in mathematical physics that consists of obtaining Taylor series for fabricated linear and nonlinear unphysical models by means of homotopy perturbation method (HPM), homotopy analysis method (HAM) and Adomian decomposition method (ADM). As an illustrative example we choose a recent application of the HPM to a dynamic system of anisotropic elasticity.

math-ph↗
arXiv2009

Harmonic oscillator in a one-dimensional box

We study a harmonic molecule confined to a one--dimensional box with impenetrable walls. We explicitly consider the symmetry of the problem for the cases of different and equal masses. We propose suitable variational functions and compare the approximate energies given by the variation method and perturbation theory with accurate numerical ones for a wide range of values of the box length. We analyze the limits of small and large box size.

quant-ph↗
arXiv2009

Variational collocation for systems of coupled anharmonic oscillators

We have applied a collocation approach to obtain the numerical solution to the stationary Schrödinger equation for systems of coupled oscillators. The dependence of the discretized Hamiltonian on scale and angle parameters is exploited to obtain optimal convergence to the exact results. A careful comparison with results taken from the literature is performed, showing the advantages of the present approach.

quant-ph↗
arXiv2009

Rational approximation to Thomas-Fermi equations

We show that a simple and straightforward rational approximation to the Thomas-Fermi equation provides the slope at origin with unprecedented accuracy and that relatively small Padé approximants are far more accurate than more elaborate approaches proposed recently by other authors. We consider both the Thomas-Fermi equation for isolated atoms and for atoms in strong magnetic fields.

math-ph↗
arXiv2008

Rayleigh-Ritz variation method and connected-moments polynomial approach

We show that the connected-moments polynomial approach proposed recently is equivalent to the well known Rayleigh-Ritz variation method in the Krylov space. We compare the latter with one of the original connected-moments methods by means of a numerical test on an anharmonic oscillator

math-ph↗
arXiv2008

On Some Perturbation Approaches to Population Dynamics

We show that the Adomian decomposition method, the time--series expansion, the homotopy--perturbation method, and the variational--iteration method completely fail to provide a reasonable description of the dynamics of the simplest prey--predator system.

math-ph↗
arXiv2008

Rational approximation to the Thomas--Fermi equation

We show that a simple and straightforward rational approximation to the Thomas--Fermi equation provides the slope at origin with unprecedented accuracy. We compare present approach with other available ones.

math-ph↗