Author Archives: Eric A. Galapon

We reconsider the problem of regularizing the divergent series $\sum_{n=1}^{\infty}n^{\alpha}$ for $\operatorname{Re}\alpha>-1$, and offer a regularization prescription that yields the Riemann zeta regularization as a special case. The development of the regularization is framed as a two-step problem. The first step is prescribing a regularization of the divergent sum $\sum_{n=1}^{\infty}n^m$ for every non-negative integer $m$;… Read More »

We consider the asymptotic evaluation of the integral transform $\int_0^\infty f(x) \, \sin^n(\lambda x)/x^n \,\text{d} x$ of an exponential type function $f(x)$ of type $\tau>0$, for large values of the parameter $\lambda$, where $n$ is a positive integer. We refer to this integral as the Sinc transform. Under the condition that $f(x)$ is even with… Read More »

The integral $\int_0^a f(t) t^{-s} \mathrm{d}t$ diverges for $\text{Re}(s) \geq \lambda + 1$, where $\lambda$ is the order of the first non-vanishing derivative of $f(t)$ at the origin. With the assumption that $f(t)$ is analytic at the origin, the finite-part of the divergent integral assumes the contour integral representation of the form $\bbint{0}{a} f(t) t^{-s}… Read More »

Previous numerical analyses on the Aharonov-Bohm (AB) operator representing the quantum free time-of-arrival (TOA) observable have indicated that its eigenfunctions represent quantum states with definite arrival time at the arrival point. Here, we give the mathematical proof that this is indeed the case. Essential to this is the consideration of the eigenfunctions of the AB… Read More »

A pair of Hermitian operators $(A,B)$ is canonical if they satisfy the commutator relation $[A,B]=i\hbar I$, where $I$ is the identity operator. However, the relation holds only in a proper subspace of the Hilbert space, referred to as the canonical domain. In infinite-dimensional Hilbert space, it was previously shown that canonical pairs exist in a… Read More »

Previous numerical analyses on the Aharonov-Bohm (AB) operator representing the quantum time-of-arrival (TOA) observable for the free particle have indicated that its eigenfunctions represent quantum states with definite arrival time at the arrival point. In this paper, we give the mathematical proof that this is indeed the case. An essential element of this proof is… Read More »