Riccati Pade Method Research Articles

Comment on: “On the computation of eigenvalues of the anharmonic Coulombic potential”

In this comment we show that the claims by Cassidy et al (J Math Chem 56: 477, 2018) about the limitations of the Riccati Pade method (RPM) are unfounded. To this end we compare the performance of the RPM and their DESCM by means of the calculation of the eigenvalues of a class of one-dimensional potentials. We show that the RPM is simple and extremely accurate. In addition to the exponential convergence of its approximate eigenvalues the RPM does not require any variable transformation, adjustable parameters or scaling that are necessary for the improvement of the performance of the DESCM. An additional remarkable feature of the RPM is that it provides upper and lower bounds to the eigenvalues of the perturbed Coulomb potentials chosen for this test.

On two different kinds of resonances in one-dimensional quantum-mechanical models

We apply the Riccati–Pade method and the Rayleigh–Ritz method with complex rotation to the study of the resonances of a one-dimensional well with two barriers. The model exhibits two different kinds of resonances and we calculate them by means of both approaches. While the Rayleigh–Ritz method reveals each set at a particular interval of rotation angles the Riccati–Pade method yields both of them as roots of the same Hankel determinants.

Accurate eigenvalues and eigenfunctions for quantum-mechanical anharmonic oscillators

The representation of the Taylor expansion of the logarithmic-derivative of the wavefunction by means of a Pade approximant, followed by an appropriate quantization condition, proves a powerful way of obtaining accurate eigenvalues of the Schrodinger equation. In this paper we investigate in detail some of the interesting features of this approach, termed Riccati-Pade method (RPM), by means of its application to anharmonic oscillators. We analyse the occurrence of many roots in the neighborhoods of the physical eigenvalues in the weak-coupling regime, and also obtain accurate coefficients of the strong-coupling expansion. We finally investigate the global and the local accuracy of the RPM eigenfunctions.

Large-order perturbation expansions for the charged harmonic oscillator

The charged oscillator defined by the Hamiltonian H=-d2/dr2+r2+ lambda /r in the domain (0, infinity ), is a particular case of the family of spike oscillators, which does not behave as a supersingular Hamiltonian. This problem is analysed around the three regions lambda to infinity , lambda to 0 and lambda to - infinity by using Rayleigh-Ritz large-order perturbative expansions. A path is found to connect the large lambda regions with the small lambda region by means of the renormalization of the series expansions in lambda . Finally, the Riccati-Pade method is used to construct an implicit expansion around lambda to 0 which extends to very large values of mod lambda mod .