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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 »

Abstract We consider the characteristic time operator introduced in Galapon (2002) [26] which is bounded and self-adjoint. For a semibounded discrete Hamiltonian with some growth condition, satisfies the canonical relation for in a dense subspace of the Hilbert space. While is not covariant, we show that it still satisfies the canonical relation in a set… Read More »

Abstract It was shown in Phys. Rev. Lett. 108, 170402 (2012) that quantum tunneling is instantaneous using a time-of-arrival (TOA) operator constructed by Weyl quantization of the classical TOA. However, there are infinitely many possible quantum images of the classical TOA, leaving it unclear if one is uniquely preferred over the others. This raises the… Read More »

Bulletin-Oct2024-print

The quantum time of arrival (TOA) problem requires the statistics of 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 »