## D.A.L. Pablico and E.A. Galapon, “Quantum traversal time across a potential well,” arXiv: 1908.03400

By | August 12, 2019

Abstract We consider the quantum traversal time of an incident wave packet across a potential well using the theory of quantum time of arrival (TOA)-operators. This is done by constructing the corresponding TOA-operator across a potential well via quantization. The expectation value of the potential well TOA-operator is compared to the free particle case for… Read More »

## “Time and Fundamentals of Quantum Mechanics,” Weizmann Institute of Science, Rehovot Israel (January 28-31, 2019)

By | January 18, 2019

## C.D. Tica and E.A. Galapon, “Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. II. Non-integer orders”, Journal of Mathematical Physics 60, 013502 (2019).

By | January 3, 2019

Get your copy here from the publisher. Compare with the original version of the paper, especially Section 3. Tica-Galapon Journal of Mathematical Physics 60, 013502 (2019) (accepted version)

## The limit to infinity: Addendum to “The problem of missing terms in term by term integration involving divergent integrals”

By | December 29, 2018

In (Galapon 2017) we considered the problem of missing terms arising from evaluating the incomplete Stieltjes transform, S_a(\omega)=\int_0^a \frac{x^{-\nu}f(x)}{(\omega+x)}\,\mathrm{d}x, \;\;\; 0

## Math hack: Souped-up integration by parts

By | December 28, 2018

Integrals are a mainstay in physics. Sooner or later we will have to evaluate an integral and when one arises its either we pick up the venerable Table of Integrals, Series, and Products by Gradhsteyn and Ryzhik (GR) or tap our keyboards and let Mathematica or Maple do the integration for us. We are lucky… Read More »

## E.A. Galapon and J.J. P. Magadan, “Quantizations of the classical time of arrival and their dynamics,” Annals of Physics 397 (2018) 278-302.

By | December 14, 2018

https://www.sciencedirect.com/science/article/pii/S0003491618302173

## An absolute beauty

By | August 26, 2017

\begin{equation*} \int_{-\infty}^{\infty} \frac{\sin\pi x}{\Gamma\!(\pi-x) \Gamma\!(\pi+x)}\, \frac{\mathrm{d}x}{x}= \frac{\pi}{\Gamma\!(\pi)^2} \end{equation*}

## C.D. Tica and E.A. Galapon, “Finite-Part Integration of the Generalized Stieltjes Transform and its dominant asymptotic behavior for small values of the parameter” arXiv:1703.07979.

By | March 24, 2017

The paper addresses the exact evaluation of the generalized Stieltjes transform $S_{\lambda}[f]=\int_0^{\infty} f(x) (\omega+x)^{-\lambda}\mathrm{d}x$ about $\omega =0$ from which the asymptotic behavior of $S_{\lambda}[f]$ for small parameters $\omega$ is directly extracted. An attempt to evaluate the integral by expanding the integrand $(\omega+x)^{-\lambda}$ about $\omega=0$ and then naively integrating the resulting infinite series term by term… Read More »

## Most downloaded: “Internal one degree of freedom is sufficient to induce exact decoherence”

By | February 26, 2017

## Galapon EA. 2017 The problem of missing terms in term by term integration involving divergent integrals. Proc. R. Soc. A 473: 20160567.

By | January 20, 2017

The operations of integration and summation cannot be interchanged arbitrarily. Some uniformity conditions must be satisfied in order for the interchange to be performed. If the conditions are not satisfied and the interchange is nevertheless carried out, some outrageous things happen. For example, the interchange may lead to an infinite series of divergent integrals; that… Read More »