arXiv subjects

Eric A. Galapon

Publications and source records attributed to Eric A. Galapon.

At least 37 records · Page 2Linked to original sources

arXiv2022

Quantum corrections to the Weyl quantization of the classical time of arrival

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 analytic potentials. In this paper, we give a full account of the said TOA-operator by explicitly solving all the terms in the expansion. We interpret the terms beyond the leading term as the quantum corrections to the Weyl quantization of the classical arrival time. These quantum corrections are expressed as some integrals of the interaction potential and their properties are investigated in detail. In particular, the quantum corrections always vanish for linear systems but nonvanishing for nonlinear systems. Finally, we consider the case of an anharmonic oscillator potential as an example.

quant-ph↗

arXiv2022

Relativistic free motion time of arrival operator for massive spin-0 particles with positive energy

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 time of arrival problem that goes beyond Razavi's original results. This allows us to use time of arrival eigenfunctions that exhibit unitary arrival to construct time of arrival distributions. The expectation value is also calculated and shown that particles can arrive earlier or later than expected classically. Lastly, the constructed time of arrival distribution, and expectation value are shown to be consistent with special relativity.

quant-ph↗

arXiv2022

Finite-part integral representation of the Riemann zeta function at odd positive integers and consequent representations

The values of the Riemann zeta function at odd positive integers, $ζ(2n+1)$, 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 $ζ(2n+1)$ are then deduced from the finite-part integral representation. Certain relations between $ζ(2n+1)$ and $ζ'(2n+1)$ are likewise deduced, from which integral representations for $ζ'(2n+1)$ are obtained.

math.NT↗

arXiv2022

Conjugates to One Particle Hamiltonians in 1-Dimension in Differential Form

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 arrival operator. Here, we are interested in finding other Hamiltonian conjugates by further studying the boundary conditions. A modified form of the time kernel equation is also considered which gives an even bigger solution space.

quant-ph↗

arXiv2021

Regularized Limit, analytic continuation and finite-part integration

Finite-part integration is a recent method of evaluating a convergent integral in terms of the finite-parts of divergent integrals deliberately induced from the convergent integral itself [E. A. Galapon, Proc. R. Soc., A 473, 20160567 (2017)]. Within the context of finite-part integration of the Stieltjes transform of functions with logarithmic growths at the origin, the relationship is established between the analytic continuation of the Mellin transform and the finite-part of the resulting divergent integral when the Mellin integral is extended beyond its strip of analyticity. It is settled that the analytic continuation and the finite-part integral coincide at the regular points of the analytic continuation. To establish the connection between the two at the isolated singularities of the analytic continuation, the concept of regularized limit is introduced to replace the usual concept of limit due to Cauchy when the later leads to a division by zero. It is then shown that the regularized limit of the analytic continuation at its isolated singularities equals the finite-part integrals at the singularities themselves. The treatment gives the exact evaluation of the Stieltjes transform in terms of finite-part integrals and yields the dominant asymptotic behavior of the transform for arbitrarily small values of the parameter in the presence of arbitrary logarithmic singularities at the origin.

math.CA↗

arXiv2020

Finite-Part Integration in the Presence of Competing Singularities: Transformation Equations for the hypergeometric functions arising from Finite-Part Integration

Finite-part integration is a recently introduced method of evaluating convergent integrals by means of the finite part of divergent integrals [E.A. Galapon, {\it Proc. R. Soc. A 473, 20160567} (2017)]. Current application of the method involves exact and asymptotic evaluation of the generalized Stieltjes transform $\int_0^a f(x)/(ω+ x)^ρ \, \mathrm{d}x$ under the assumption that the extension of $f(x)$ in the complex plane is entire. In this paper, the method is elaborated further and extended to accommodate the presence of competing singularities of the complex extension of $f(x)$. Finite part integration is then applied to derive consequences of known Stieltjes integral representations of the Gauss function and the generalized hypergeometric function which involve Stieltjes transforms of functions with complex extensions having singularities in the complex plane. Transformation equations for the Gauss function are obtained from which known transformation equations are shown to follow. Also, building on the results for the Gauss function, transformation equations involving the generalized hypergeometric function $\,_3F_2$ are derived.

math.CA↗

arXiv2019

Quantum Traversal Time Across a Potential Well

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 the same incident wave packet. The comparison yields a closed-form expression of the quantum well traversal time which explicitly shows the classical contributions of the positive and negative momentum components of the incident wave packet and a purely quantum mechanical contribution significantly dependent on the well depth. An incident Gaussian wave packet is then used as an example. It is shown that for shallow potential wells, the quantum well traversal time approaches the classical traversal time across the well region when the incident wave packet is spatially broad and approaches the expected quantum free particle traversal time when the wave packet is localized. For deep potential wells, the quantum traversal time oscillates from positive to negative implying that the wave packet can be advanced or delayed.

quant-ph↗

arXiv2019

Exact Decoherence and Orthogonal Pointer States Brought by One Degree of Freedom: von Neumann Equation Approach and Examples

In a quantum measurement setting, it is known that environment-induced decoherence theory describes the emergence of effectively classical features of the quantum system-measuring apparatus composite system when the apparatus is allowed to interact with the environment. In [E.A. Galapon {\it EPL} {\bf 113} 60007 (2016)], a measurement model is found to have the feature of inducing exact decoherence at a finite time via one internal degree of freedom of the apparatus provided that the apparatus is decomposed into a pointer and an inaccessible probe, with the pointer and the probe being in momentum-limited initial states. However, an issue can be raised against the model: while the factorization method of the time evolution operator used there is formally correct, it is not completely rigorous due to some unstated conditions on the validity of the factorization in the Hilbert space of the model. Furthermore, no examples were presented there in implementing the measurement scheme in specific quantum systems. The goal of this paper is to re-examine the model and confirm its features independently by solving the von Neumann equation for the joint state of the composite system as a function of time. This approach reproduces the joint state obtained in the original work, leading to the same conditions for exact decoherence and orthogonal pointer states when the required initial conditions on the probe and pointer are imposed. We illustrate the exact decoherence process in the measurement of observables of a spin-1/2 particle and a quantum harmonic oscillator by using the model.

quant-ph↗

arXiv2019

Quantum free fall motion and quantum violation of weak equivalence principle

The weak equivalence principle (WEP) in the quantum regime has been the subject of many studies with a broad range of approach to the problem. Here we tackle the problem anew through the time of arrival (TOA) operator approach by constructing the time of arrival operator for a non-relativistic and structureless particle that is projected upward in a uniform gravitational field with an intended arrival point below the classical turning point. The TOA-operator is constructed under the constraint that the inertial and gravitational masses are equivalent, and that Galilean invariance is preserved. These constraints are implemented by Weyl-quantization of the corresponding classical time of arrival function for the projectile. The expectation value of the TOA-operator is explicitly shown to be equal to the classical time of arrival plus mass-dependent quantum correction terms, implying incompatibility of the weak equivalence principle with quantum mechanics. The full extent of the violation of the WEP is shown through the mass dependence of time of arrival distribution for the projectile.

quant-ph↗

arXiv2018

Finite-Part Integration of the Generalized Stieltjes Transform and its dominant asymptotic behavior for small values of the parameter. II. Non-integer orders

The paper constitutes the second part on the subject of finite part integration of the generalized Stieltjes transform $S_λ[f]=\int_0^{\infty} f(x) (ω+x)^{-λ}\mathrm{d}x$ about $ω= 0$ where now $λ$ is a non-integer positive real number. Divergent integrals with singularities at the origin are induced by writing $(ω+x)^{-λ}$ as a binomial expansion about $ω= 0$ and interchanging the order of operations of integration and summation. The prescription of finite part integration is then implemented by interpreting these divergent integrals as finite part integrals which are rigorously represented as complex contour integrals. The same contour is then used to express $S_λ[f]$ itself as a complex contour integral. This led to the recovery of the terms missed by naive term-wise integration which themselves are finite parts of divergent integrals whose singularity is at the finite upper limit of integration. When the function $f(x)$ has a zero at the origin of order $m=0,1,\dots$ such that $m-λ<0$ the correction terms missed out by naive term by term integration give the dominant contribution to $S_λ[f]$ as $ω\to 0$. Otherwise, the correction term is sub-dominant to the leading convergent terms in the naive term by term integration. We apply these results by obtaining exact and asymptotic representations of the Kummer and Gauss hypergeometric functions by evaluating their known Stieltjes integral representations. We then apply the method of finite part integration to obtain the asymptotic behavior of a generalization of the Stieltjes integral which is relevant in the calculation of the effective index of refraction of a shallow potential well.

math-ph↗

arXiv2018

The problem of missing terms in term by term integration involving divergent integrals

Term by term integration may lead to divergent integrals, and naive evaluation of them by means of, say, analytic continuation or by regularization or by the finite part integral may lead to missing terms. Here, under certain analyticity condition, the problem of missing terms for the incomplete Stieltjes transform, $\int_0^a f(x) (ω+x)^{-1} \mathrm{d}x$, and the Stieltjes transform itself, $\int_0^{\infty} f(x) (ω+x)^{-1} \mathrm{d}x$, is resolved by lifting the integration in the complex plane. It is shown that the missing terms arise from the singularities of the complex valued function $f(z) (ω+ z)^{-1}$, with the divergent integrals arising from term by term integration interpreted as finite part integrals.

math-ph↗

arXiv2018

Integration by divergent integrals: Calculus of divergent integrals in term by term integration

It is known in the case of the Stieltjes transform that evaluating the integral by expanding the kernel of transformation followed by term by term integration leads to an infinite series of divergent integrals. Moreover, it is known that merely assigning values to the divergent integrals leads to missing terms. This problem of missing terms was earlier resolved in the complex plane, which required the divergent integrals to assume values equal to their finite parts under a set of rules or calculus governing the use of the finite part integral [E.A. Galapon, {\it Proc. Roy. Soc. A} {\bf 473}, 20160567 (2017)]. However, divergent integrals assume a spectrum of values arising from the different ways of removing the divergences in the integration. Here we show that the other interpretations of divergent integrals follow the same set of rules as those of the finite part integral, and then formulate the calculus of divergent integrals arising from term-by-term integration. We find that the calculus does not prescribe a specific interpretation to the divergent integrals but allows us to assign arbitrary values to them, with the missing terms automatically emerging from the chosen interpretation of the divergent integrals itself.

math-ph↗

arXiv2018

Finite-Part Integration of the Generalized Stieltjes Transform and its dominant asymptotic behavior for small values of the parameter Part I: Integer orders

The paper addresses the exact evaluation of the generalized Stieltjes transform $S_{n}[f]=\int_0^{\infty} f(x) (ω+x)^{-n}\mathrm{d}x$ of integral order $n=1,2, 3,\dots$ about $ω=0$ from which the asymptotic behavior of $S_{n}[f]$ for small parameters $ω$ is directly extracted. An attempt to evaluate the integral by expanding the integrand $(ω+x)^{-n}$ about $ω=0$ and then naively integrating the resulting infinite series term by term lead to an infinite series whose terms are divergent integrals. Assigning values to the divergent integrals, say, by analytic continuation or by Hadamard's finite part is known to reproduce only some of the correct terms of the expansion but completely misses out a group of terms. Here we evaluate explicitly the generalized Stieltjes transform by means of finite-part integration recently introduced in [E.A. Galapon, {\it Proc. Roy. Soc. A} {\bf 473}, 20160567 (2017)]. It is shown that, when $f(x)$ does not vanish or has zero of order $m$ at the origin such that $(n-m)\geq 1$, the dominant terms of $S_{n}[f]$ as $ω\rightarrow 0$ come from contributions arising from the poles and branch points of the complex valued function $f(z) (ω+z)^{-n}$. These dominant terms are precisely the terms missed out by naive term by term integration. Furthermore, it is demonstrated how finite-part integration leads to new series representations of special functions by exploiting their known Stieltjes integral representations. Finally, the application of finite part integration in obtaining asymptotic expansions of the effective diffusivity in the limit of high Peclet number, the Green-Kubo formula for the self-diffusion coefficient and the antisymmetric part of the diffusion tensor in the weak noise limit is discussed.

math-ph↗

arXiv2018

Quantizations of the classical time of arrival and their dynamics

The classical time of arrival in the interacting case is quantized by way of quantizing its expansion about the free time of arrival. The quantization is formulated in coordinate representation which represents ordering rules in terms of two variable polynomial functions. This leads to representations of the quantized time of arrival operators as integral operators whose kernels are determined by the chosen ordering rule. The formulation lends itself to generalization which allows construction of time of arrival operators that cannot be obtained by direct quantization using particular ordering rules. Wey, symmetric and Born-Jordan quantizations are specifically studied. The dynamics of the eigenfunctions of the different time of arrival operators are investigated. The eigenfunctions exhibit unitary arrival at the intended arrival point at their respective eigenvalues.

quant-ph↗

arXiv2016

Synchronizing quantum and classical clocks made of quantum particles

We demonstrate that the quantum corrections to the classical arrival time for a quantum object in a potential free region of space, as computed by Galapon [Phys. Rev. A {\bf 80}, 030102(R) (2009)], can be eliminated up to a given order of $\hbar$ by choosing an appropriate position-dependent phase for the object's wavefunction. This then implies that we can make the quantum arrival time of the object as close as possible to its corresponding classical arrival time, allowing us to synchronize a classical and quantum clock which tells time using the classical and quantum arrival time of the object, respectively. We provide an example for synchronizing such a clock by making use of a quantum object with a position-dependent phase imprinted on the object's initial wavefunction with the use of an impulsive potential.

quant-ph↗

arXiv2016

The Cauchy Principal Value and the finite part integral as values of absolutely convergent integrals

The divergent integral $\int_a^b f(x)(x-x_0)^{-n-1}\mathrm{d}x$, for $-\infty<a<x_0<b<\infty$ and $n=0, 1, 2, \dots$, is assigned, under certain conditions, the value equal to the simple average of the contour integrals $\int_{C^{\pm}} f(z)(z-x_0)^{-n-1}\mathrm{d}z$, where $C^+$ ($C^-$) is a path that starts from $a$ and ends at $b$, and which passes above (below) the pole at $x_0$. It is shown that this value, which we refer to as the Analytic Principal Value, is equal to the Cauchy principal value for $n=0$ and to the finite-part of the divergent integral for positive integer $n$. This implies that, where the conditions apply, the Cauchy principal value and the finite-part integral are in fact values of absolutely convergent integrals. Moreover, it leads to the replacement of the boundary values in the Sokhotski-Plemelj-Fox Theorem with integrals along some arbitrary paths. The utility of the Analytic Principal Value in the numerical, analytical and asymptotic evaluation of the Cauchy principal value and the finite-part integral is discussed and demonstrated.

math-ph↗

arXiv2015

Mixture of entangled pure states with maximally mixed one-qudit reduced density matrices

We study the conditions when mixtures of entangled pure states with maximally mixed one-qudit reduced density matrices remain entangled. We found that the resulting mixed state remains entangled when the number of entangled pure states to be mixed is less than or equal to the dimension of the pure states. For the latter case of mixing a number of pure states equal to their dimension, we found that the mixed state is entangled provided that the entangled pure states to be mixed are not equally weighted. We also found that one can restrict the set of pure states that one can mix from in order to ensure that the resulting mixed state is genuinely entangled.

quant-ph↗

arXiv2015

Exactly decohering quantum measurement without environment

Current quantum orthodoxy claims that the statistical collapse of the wave-function arises from the interaction of the measuring instrument with its environment through the phenomenon known as environment induced decoherence. Here it is shown that there exists a measurement scheme that is exactly decohering without the aid of an environment. The scheme relies on the assumption that the meter is decomposable into probe and pointer, with the probe taken to be inaccessible for observation. Under the assumption that the probe and the pointer initial states are momentum limited, it is shown that coherences die out within a finite interval of time and the pointer states are exactly orthogonal. These lead to the fundamental realization that dispersion of correlation does not require an external infinite number of degrees of freedom. An internal one degree of freedom is already sufficient to delocalize the correlations and leave its subparts in a classical mixed state, so that decoherence may occur even for isolated measuring instruments.

quant-ph↗