The 1958 Pekeris-Accad-WEIZAC Ground-Breaking Collaboration that Computed Ground States of Two-Electron Atoms (and its 2010 Redux)

In order to appreciate how well off we mathematicians and scientists are today, with extremely fast hardware and lots and lots of memory, as well as with powerful software, both for numeric and symbolic computation, it may be a good idea to go back to the early days of electronic computers and compare how things went then. We have chosen, as a case study, a problem that was considered a huge challenge at the time. Namely, we looked at C.L. Pekeris's seminal 1958 work on the ground state energies of two-electron atoms. We went through all the computations ab initio with today's software and hardware, with a special emphasis on the symbolic computations which in 1958 had to be made by hand, and which nowadays can be automated and generalized.Comment: 8 pages, 2 photos, final version as it appeared in the journa

Rotational Correction on the Morse Potential Through the Pekeris Approximation and Nikiforov-Uvarov Method

The Nikiforov-Uvarov method is employed to calculate the the Schrodinger equation with a rotation Morse potential. The bound state energy eigenvalues and the corresponding eigenfunction are obtained. All of these calculation present an effective and clear method under a Pekeris approximation to solve a rotation Morse model. Meanwhile the results got here are in a good agreement with ones before.Comment: 11 pages, no figure, submitted to Chemical Physics Letters, (2005

Quantum properties of the electron field in Kerr-Newman black hole manifolds

We study some spectral features of the one-particle electron Hamiltonian obtained by separating the Dirac equation in a Kerr-Newman black hole background. We find that the essential spectrum includes the whole real line. As a consequence, there is no gap in the spectrum and discrete eigenvalues are not allowed for any value of the black hole charge $Q$ and angular momentum $J$. Our spectral analysis will be also related to the dissipation of the black hole angular momentum and charge.Comment: 9 pages, uses revte

A note on dynamic surface displacements in an elastic half-space

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41710/1/707_2005_Article_BF01260886.pd

Improved analytical approximation to arbitrary l-state solutions of the Schrodinger equation for the hyperbolical potentials

A new approximation scheme to the centrifugal term is proposed to obtain the $l\neq 0$ bound-state solutions of the Schr\"{o}dinger equation for an exponential-type potential in the framework of the hypergeometric method. The corresponding normalized wave functions are also found in terms of the Jacobi polynomials. To show the accuracy of the new proposed approximation scheme, we calculate the energy eigenvalues numerically for arbitrary quantum numbers $n$ and $l$ with two different values of the potential parameter $\sigma_{\text{0}}.$ Our numerical results are of high accuracy like the other numerical results obtained by using program based on a numerical integration procedure for short-range and long-range potentials. The energy bound-state solutions for the s-wave ($l=0$) and $\sigma_{0}=1$ cases are given.Comment: 17 page

An Acoustic Emission Evaluation of Environmentally Assisted Cracking of 7039-T6 Aluminum

Environmentally assisted cracking (EAC) is a significant problem in modern structures. The combination of a susceptible material, an adverse environment and mechanical stress can lead to unexpected failure of a structure by catastrophic crack growth. The mid-air failure of the aluminum alloy bulkhead and the subsequent loss of life on a Aloha Airlines flight on April 28, 1988 as shown in figure 1, illustrates this fact. Additionally, the operating environment of the US Army contributes to premature failure of structures such as aluminum alloy armor, high strength steel armor and high strength steel control components on Army helicopters [1]. These failures not only endanger life but they also seriously hamper the fighting readiness of U.S. forces because of equipment down time for inspection and repair of faulty components. Work has been performed to better characterize EAC resistance in high strength aluminum armor alloys [2]. These high strength alloys are particularly prone to failure in a chloride environment, an environment encountered in most of the world. If we plan to avoid such failures, we must better understand the EAC phenomena and more diligently detect growing cracks before they become critical in length. One characterization technique that promises to serve well both as a laboratory tool for understanding EAC and as a field device for detecting EAC is acoustic emission evaluation

On the evaluation of some three-body variational integrals

Stable recursive relations are presented for the numerical computation of the integrals $$\int d{\bf r}_1 d{\bf r}_2 r_1^{l-1} r_2^{m-1} r_{12}^{n-1} \exp{\{-\alpha r_1 -\beta r_2 -\gamma r_{12}\}}$$ ($l$, $m$ and $n$ integer, $\alpha$, $\beta$ and $\gamma$ real) when the indices $l$, $m$ or $n$ are negative. Useful formulas are given for particular values of the parameters $\alpha$, $\beta$ and $\gamma$.Comment: 12 pages, 1 figure (PS) and 3 tables. Old figures 2 and 3 replaced by Tables I and III. A further table added. Paper enlarged giving some tips on the convergence of quadrature