This is a major revision of the original preprint. The one-sided and full Hilbert transforms are evaluated exactly by means of the method of finite-part integration [E.A. Galapon, Proc. Roy. Soc. A \textbf{473}, 20160567 (2017)]. In general, the result consists of two terms—the first is an infinite series of finite-part of divergent integrals, and the… Read More »
The quantum time of arrival (TOA) problem requires statistics of the measured arrival times given only the initial state of a particle. Following the standard framework of quantum theory, the problem translates into finding an appropriate quantum image of the classical arrival time $\mathcal{T}_C(q,p)$, usually in operator form $\hat{\mathrm{T}}$. In this paper, we consider the… Read More »
A prescription is devised to utilize a novel convergent expansion in the strong-asymptotic regime of the Stieltjes integral and its generalizations [Galapon E.A Proc.R.Soc A 473, 20160567(2017)] to sum the associated divergent series of Stieltjes across all asymptotic regimes. The novel expansion makes use of the divergent negative-power moments which were treated as Hadamard’s finite… Read More »
Finite-part integration is a recently introduced method of evaluating well-defined integrals using the finite-part of divergent integrals [E.A. Galapon, Proc. Roy. Soc. A \textbf{473}, 20160567 (2017)]. Here finite-part integration is applied in the exact evaluation of the one-sided, $\operatorname{PV}\!\int_0^{\infty} h(x) (\omega-x)^{-1}\mathrm{d}x$, and full, $\mathrm{PV}\!\int_{-\infty}^{\infty}h(x) (\omega-x)^{-1}\mathrm{d}x$, Hilbert transforms, together with the reductions of the later transform… Read More »
The tunneling time problem earlier studied in Phys. Rev. Lett. \textbf{108}, 170402 (2012) using a non-relativistic time-of-arrival (TOA) operator predicted that tunneling time is instantaneous implying that the wavepacket becomes superluminal below the barrier. The non-relativistic treatment raises the question whether the superluminal behavior is a mere non-relativistic phenomenon or an an inherent quantum effect… Read More »
A time of arrival (TOA) operator that is conjugate with the system Hamiltonian was constructed by Galapon without canonical quantization in [J. Math. Phys. \textbf{45}, 3180 (2004)]. The constructed operator was expressed as an infinite series but only the leading term was investigated which was shown to be equal to the Weyl-quantized TOA-operator for entire… Read More »
Theorem 1. The reciprocal of $\pi$ assumes the following family of integral representations \begin{equation} \frac{1}{\pi}= \frac{\left[\prod_{k=1}^{m+n}(2k-1)\right]^2 \lambda^{2m+1}}{(2m)!\, 2^{4m+2n}} \int_0^{\infty} \frac{x^{2m} J_{2m+1}(\lambda x)}{\Gamma\left(m+ n+\frac{1}{2} + x\right) \Gamma\left(m+n+\frac{1}{2}-x\right)}\,{\rm d}x \end{equation} \begin{equation} \frac{1}{\pi}= \frac{\left[\prod_{k=1}^{m+n}(2k-1)\right]^2 \lambda^{2m+2}}{(2m+1)!\, 2^{4m+2n+1}} \int_0^{\infty} \frac{x^{2m+1} J_{2m+2}(\lambda x)}{\Gamma\left(m+n+ \frac{1}{2} + x\right) \Gamma\left(m+n+\frac{1}{2}-x\right)}\,{\rm d}x \end{equation} for $m=0, 1, 2, \dots$, $n=1, 2, 3,\dots$ and $\lambda\geq\pi$, where $J_n(x)$… Read More »
The values of the Riemann zeta function at odd positive integers, $\zeta(2n+1)$, are shown to admit a representation proportional to the finite-part of the divergent integral $\int_0^{\infty} t^{-2n-1} \operatorname{csch}t\,\mathrm{d}t$. Integral representations for $\zeta(2n+1)$ are then deduced from the finite-part integral representation. Certain relations between $\zeta(2n+1)$ and $\zeta'(2n+1)$ are likewise deduced, from which integral representations for… Read More »
A relativistic version of the Aharonov-Bohm time of arrival operator for spin-0 particles was constructed by Razavi in [Il Nuovo Cimento B \textbf{63}, 271 (1969)]. We study the operator in detail by taking its rigged Hilbert space extension. It is shown that the rigged Hilbert space extension of the operator provides more insights into the… Read More »
A time operator is a Hermitian operator that is canonically conjugate to a given Hamiltonian. For a particle in 1-dimension, a Hamiltonian conjugate operator in position representation can be obtained by solving a hyperbolic second-order partial differential equation, known as the time kernel equation, with some boundary conditions. One possible solution is the time of… Read More »