arXiv subjects

Filippo Giraldi

Publications and source records attributed to Filippo Giraldi.

At least 19 recordsLinked to original sources

arXiv2025

On the Laplace transforms of derivatives of special functions with respect to parameters

This article is devoted to derivation of the Laplace transforms of the derivatives with respect to parameters of certain special functions, namely, the Mittag-Leffler type, Wright and Le Roy type functions. These formulas show interconnection of these functions and lead to better understanding of their behaviour on the real line. These formulas are represented in the convoluted form and reconstructed in a more suitable form by using Efros theorem

math.GM↗

arXiv2025

A class of positive Fox H-functions

The Fox $H$-function is a special function which is defined via the Mellin-Barnes integrals and produces, as particular cases, Wright generalized hypergeometric functions, MacRobert's $E$-functions and Meijer $G$-functions, to name but few. Various cases of non-negative Fox $H$-functions are obtained in literature by relying on the properties of integral transforms and the complete monotonicity. In the present scenario, Fox $H$-functions, which are positive on $\mathbb{R}^+$, are determined via the Mellin convolution products of finite combinations, with possible repetitions, of elementary functions. The chosen elementary functions are non-negative on $\mathbb{R}^+$ and are defined via stretched exponential and power laws. Further forms of positive Fox $H$-functions can be obtained from the former via elementary properties and integral transforms. As particular cases, we determine forms of Wright generalized hypergeometric functions, MacRobert's $E$-functions and Meijer $G$-functions which are positive on $\mathbb{R}^+$.

math.CV↗

arXiv2024

On differentiation with respect to parameters of the functions of the Mittag-Leffler type

The formal term-by-term differentiation with respect to parameters is demonstrated to be legitimate for the Mittag-Leffler type functions. The justification of differentiation formulas is made by using the concept of the uniform convergence. This approach is applied to the Mittag-Leffler function depending on two parameters and, additionally, for the $3$-parametric Mittag-Leffler functions (namely, for the Prabhakar function and the Le Roy type functions), as well as for the $4$-parametric Mittag-Leffler function (and, in particular, for the Wright function). The differentiation with respect to the involved parameters is discussed also in case those special functions which are represented via the Mellin-Barnes integrals.

math.GM↗

arXiv2022

Truncated generalized coherent states

A generalization of the canonical coherent states of a quantum harmonic oscillator has been performed by requiring the conditions of normalizability, continuity in the label and resolution of the identity operator with a positive weight function. Relying on this approach, in the present scenario coherent states are generalized over the canonical or finite dimensional Fock space of the harmonic oscillator. A class of generalized coherent states is determined such that the distribution of the number of excitations departs from the Poisson statistics according to combinations of stretched exponential decays, power laws and logarithmic forms. The analysis of the Mandel parameter shows that these generalized coherent states exhibit (non-classical) sub-Poissonian or super-Poissonian statistics of the number of excitations for small values of the label, according to determined properties. The statistics is uniquely sub-Poissonian for large values of the label. As particular cases, truncated Wright generalized coherent states exhibit uniquely non-classical properties, differently from the truncated Mittag-Leffler generalized coherent states.

quant-ph↗

arXiv2021

Short-time coherence of a qubit and measurement apparatus

The effects of the measurement apparatus on quantum coherence are studied by considering a purely dephasing model of a qubit. The initial state is prepared from a thermal state of the whole system by performing a nonselective measurement on the qubit. The magnitude of the initial postmeasurement coherence is bounded by the value $1/2$, which is realized with special measurement schemes and in the low-temperature limit. The magnitude of coherence identically vanishes, increases or decreases with approximately constant velocity over a determined short time scale, according to the choice of the preparation measurement. The maximization of the short-time increasing or decreasing velocity is favored by the choice of further special measurement schemes and the high-temperature limit. The measurement apparatus allows to manipulate quantum coherence of the qubit over short times via nonselective preparation measurements.

quant-ph↗

arXiv2020

The Riemann hypothesis via the Mellin transform, power series and the reflection relations

A proof of the Riemann hypothesis is proposed by relying on the properties of the Mellin transform. The function $\mathfrak{G}_{\eta}\left(t\right)$ is defined on the set $\bar{\mathbb{R}}_+$ of the non-negative real numbers, in term of a special power series, in such a way that the Mellin transform $\hat{\mathfrak{G}}_{\eta}\left(s\right)$ of the function $\mathfrak{G}_{\eta}\left(t\right)$ does not vanish in the fundamental strip $0<\operatorname{Re} s <1/2$. In this strip every zero of the Riemann zeta function $\zeta\left(1-s\right)$ is a zero of the function $\hat{\mathfrak{G}}_{\eta}\left(s\right)$. Consequently, it is proved that no zero of the Riemann zeta function $\zeta\left(s\right)$ exists in the strip $1/2<\operatorname{Re} s <1$. The reflection relations, which hold around the line $\operatorname{Re} s =1/2$ for $s\neq 0,1$, prove that no zero of the Riemann zeta function $\zeta\left(s\right)$ exists in the strip $0<\operatorname{Re} s<1/2$. In conclusion, it is proved that no zero of the Riemann zeta function $\zeta\left(s\right)$ exists in the strip $0<\operatorname{Re} s<1$ for $\operatorname{Re} s\neq 1/2$.

math.GM↗

arXiv2019

Regularities in the transformation of the oscillating decay rate in moving unstable quantum systems

Decay laws of moving unstable quantum systems with oscillating decay rates are analyzed over intermediate times. The transformations of the decay laws at rest and of the intermediate times at rest, which are induced by the change of reference frame, are obtained by decomposing the modulus of the survival amplitude at rest into purely exponential and exponentially damped oscillating modes. The mass distribution density is considered to be approximately symmetric with respect to the mass of resonance. Under determined conditions, the modal decay widths at rest, $\Gamma_j$, and the modal frequencies of oscillations at rest, $\Omega_j$, reduce regularly, $\Gamma_j/\gamma$ and $\Omega_j/\gamma$, in the laboratory reference frame. Consequently, the survival probability at rest, the intermediate times at rest and, if the oscillations are periodic, the period of the oscillations at rest transform regularly in the laboratory reference frame according to the same time scaling, over a determined time window. The time scaling reproduces the relativistic dilation of times if the mass of resonance is considered to be the effective mass at rest of the moving unstable quantum system with relativistic Lorentz factor $\gamma$.

quant-ph↗

arXiv2019

Transformation of intermediate times in the decays of moving unstable quantum systems via the exponential modes

The transformation of canonical decay laws of moving unstable quantum systems is studied by approximating, over intermediate times, the decay laws at rest with superpositions of exponential modes via the Prony analysis. The survival probability $\mathcal{P}_p(t)$, which is detected in the laboratory reference frame where the unstable system moves with constant linear momentum $p$, is represented by the transformed form $\mathcal{P}_0\left(\varphi_p(t)\right)$ of the survival probability at rest $\mathcal{P}_0(t)$. The transformation of the intermediate times, which is induced by the change of reference frame, is obtained by evaluating the function $\varphi_p(t)$. Under determined conditions, this function grows linearly and the survival probability transforms, approximately, according to a scaling law over an estimated time window. The relativistic dilation of times holds, approximately, over the time window if the mass of resonance of the mass distribution density is considered to be the effective mass at rest of the moving unstable quantum system.

quant-ph↗

arXiv2018

Time dilation in the oscillating decay laws of moving two-mass unstable quantum states

The decay of a moving system is studied in case the system is initially prepared in a two-mass unstable quantum state. The survival probability $\mathcal{P}_p(t)$ is evaluated over short and long times in the reference frame where the unstable system moves with constant linear momentum $p$. The mass distribution densities of the two mass states are tailored as power laws with powers $α_1$ and $α_2$ near the non-vanishing lower bounds $μ_{0,1}$ and $μ_{0,2}$ of the mass spectra, respectively. If the powers $α_1$ and $α_2$ differ, the long-time survival probability $\mathcal{P}_p(t)$ exhibits a dominant inverse-power-law decay and is approximately related to the survival probability at rest $\mathcal{P}_0(t)$ by a time dilation. The corresponding scaling factor $χ_{p,k}$ reads $\sqrt{1+p^2/μ_{0,k}^2}$, the power $α_k$ being the lower of the powers $α_1$ and $α_2$. If the two powers coincide and the lower bounds $μ_{0,1}$ and $μ_{0,2}$ differ, the scaling relation is lost and damped oscillations of the survival probability $\mathcal{P}_p(t)$ appear over long times. By changing reference frame, the period $T_0$ of the oscillations at rest transforms in the longer period $T_p$ according to a factor which is the weighted mean of the scaling factors of each mass, with non-normalized weights $μ_{0,1}$ and $μ_{0,2}$.

quant-ph↗

arXiv2018

Time dilation in relativistic quantum decay laws of moving unstable particles

The relativistic quantum decay laws of moving unstable particles are analyzed for a general class of mass distribution densities which behave as power laws near the (non-vanishing) lower bound $μ_0$ of the mass spectrum. The survival probability $\mathcal{P}_p(t)$, the instantaneous mass $M_p(t)$ and the instantaneous decay rate $Γ_p(t)$ of the moving unstable particle are evaluated over short and long times for an arbitrary value $p$ of the (constant) linear momentum. The ultrarelativistic and non-relativistic limits are studied. Over long times, the survival probability $\mathcal{P}_p(t)$ is approximately related to the survival probability at rest $\mathcal{P}_0(t)$ by a scaling law. The scaling law can be interpreted as the effect of the relativistic time dilation if the asymptotic value $M_p\left(\infty\right)$ of the instantaneous mass is considered as the effective mass of the unstable particle over long times. The effective mass has magnitude $μ_0$ at rest and moves with linear momentum $p$ or, equivalently, with constant velocity $1\Big/\sqrt{1+μ_0^2\big/p^2}$. The instantaneous decay rate $Γ_p(t)$ is approximately independent of the linear momentum $p$, over long times, and, consequently, is approximately invariant by changing reference frame.

quant-ph↗

arXiv2017

Damped oscillations of the energy of a bosonic bath due to spectral gaps and special initial correlations

The energy of the bosonic bath and the flow of quantum information are analyzed over short and long times in local dephasing channels for special correlated or factorized initial conditions, respectively, which involve thermal states. The continuous distribution of frequency modes of the bosonic bath exhibits a spectral gap over low frequencies. The bath energy shows oscillatory behaviors around the asymptotic value and information is alternatively lost and gained by the open system. Due to the low-frequency gap, the damped oscillations become regular over long times and the frequency of the oscillations coincides with the upper cut-off frequency of the spectral gap. Sequences of long-time intervals are obtained over which the bath energy increases (decreases), for the correlated initial conditions, and information is lost (gained) by the open system, for the factorized initial configurations, even at different temperatures. Such long-time correspondence between the variations of the bath energy and of the information is reversed if compared to the one obtained without the low-frequency gap. The correspondence fails if the spectral density is tailored according to power laws with odd natural powers near the upper cut-off frequency of the spectral gap.

quant-ph↗

arXiv2017

Sequences of information backflow in local dephasing channels with spectral gaps

The flow of quantum information in local dephasing channels is analyzed over short and long times in case the structured reservoirs of frequency modes exhibit a spectral gap in the density of modes over low frequencies. The presence of the low-frequency gap with upper cut-off frequency $ω_g$ produces over the time scale $1/ω_g$ an infinite sequence of time intervals over which information backflow appears. Such time intervals are generally irregular but, under certain conditions, exhibit the following bounds: the $n$th backflow has certainly started at the instant $π\left(1+2(n-1)\right)/ω_g$, and certainly ended at the instant $2πn/ω_g$, for every $n=1,2,\ldots$. The intervals become regular over long times, tend to the asymptotic length $π/ω_g$ as supremum value, and are described analytically in terms of the structure of the spectral density near the cut-off frequency. Consequently, engineering structured reservoirs of frequency modes with low-frequency spectral gaps produces in local dephasing channels regular and controllable sequences of information backflow and recoherence over long times, along with non-Markovian evolution.

quant-ph↗

arXiv2017

Bath energy for correlated initial states versus information flow in local dephasing channels

Variations of the bath energy are compared with the information flow in local dephasing channels. Special correlated initial conditions are prepared from the thermal equilibrium of the whole system, by performing a selective measurement on the qubit. The spectral densities under study are ohmic-like at low frequencies and include logarithmic perturbations of the power-law profiles. The bath and the correlation energy alternately increase or decrease, monotonically, over long times, according to the value of the ohmicity parameter, following logarithmic and power laws. Consider initial conditions such that the environment is in a thermal state, factorized from the state of the qubit. In the super-ohmic regime the long-time features of the information flow are transferred to the bath and correlation energy, by changing the initial condition from the factorized to the specially correlated, even with different temperatures. In fact, the low-frequency structures of the spectral density that provide information backflow with the factorized initial condition, induce increasing (decreasing) bath (correlation) energy with the specially correlated initial configuration. By performing the same change of initial conditions, the spectral properties providing information loss, produce decrease (increase) of the bath (correlation) energy.

quant-ph↗

arXiv2016

Bath correlation functions for logarithmic spectral densities

We study the bath correlation functions (BCFs) of open quantum systems interacting with thermal baths, in case the spectral densities (SDs) exhibit removable logarithmic singularities at low frequencies and are arbitrarily shaped at higher frequencies. The singularities consist in arbitrarily positive or negative powers of logarithmic functions, as additional factors for the power laws of the Ohmic-like SDs. If the SD vanishes sufficiently fast at high frequencies the short time behavior of the BCF is algebraic. The long time behavior of the BCF exhibits a variety of relaxations that involve inverse power laws and arbitrary powers of logarithmic forms. The imaginary part of the BCF shows over long times regular dependence on the low frequency structure of the SD, except for certain conditions where the ohmicity parameter takes odd natural values. Same dependence holds for the real part of the BCF at non-vanishing temperatures. At zero temperature the real part of the BCF exhibits over long times the same regular relationship with the low frequency structure of the SD, except for certain conditions involving even natural values of the ohmicity parameter. The exceptional conditions provide relaxations that are faster than those obtained via the regular dependence. In this way, various long time relaxations of the BCF that are slower than exponential decays and arbitrarily faster or slower than inverse power laws, can be interpreted in terms of removable logarithmic singularities in the low frequency structure SD of an open quantum system.

quant-ph↗

arXiv2016

Regular patterns in the information flow of local dephasing channels

Consider local dephasing processes of a qubit that interacts with a structured reservoir of frequency modes or a thermal bath, with Ohmic-like spectral density (SD). It is known that non-Markovian evolution appears uniquely above a temperature-dependent critical value of the Ohmicity parameter, and non-Markovianity can be induced by properly engineering the external environment. In the same scenario, we find that the flow of quantum information shows regular patterns: alternate directions appear in correspondence of periodical intervals of the Ohmicity parameter $α_0$. The information flows back into the system over long times for $2+4n<α_0<4+4n$, at zero temperature, and for $3+4n<α_0<5+4n$, at non-vanishing temperatures, where $n=0,1,2,\ldots$. Otherwise, the long time information flows into the environment. In the transition from vanishing to arbitrary non-vanishing temperature, the long time back-flow of information is stable for $3+4n<α_0<4+4n$, while it is reverted for $2+4n<α_0<3+4n$ and $4+4n<α_0<5+4n$. The patterns of the information flow are not altered if the low frequency Ohmic-like profiles of the SDs are perturbed with additional factors that consist in arbitrary powers of logarithmic forms. Consequently, the flow of information can be controlled, directed and reverted over long times by engineering a wide variety of reservoirs that includes and continuously departs from the Ohmic-like structure at low frequencies. Non-Markovianity and recoherence appear according to the same rules along with the back-flow of information.

quant-ph↗

arXiv2015

Some transcendental equations on the Stieltjes cone

A general class of transcendental equations in complex domain is considered for functions belonging to the Stieltjes cone. Under certain conditions each transcendental equation has no solution or one, at most, in the complex plane cut along the negative real axis. The unique solution is real valued and positive with an analytical bound. Particular cases consist in transcendental equations containing exponential, hyperbolic, power law, logarithmic and special functions. The present approach provides a simple way to prove that some special functions have no zero in certain sectors of the complex plane cut along the negative real axis.

math.CV↗

arXiv2013

Open system approach to the internal dynamics of a model multilevel molecule

A model multilevel molecule described by two sets of rotational internal energy levels of different parity and degenerate ground states, coupled by a constant interaction, is considered, by assuming that the random collisions in a gas of identical molecules, provoke transitions between adjacent energy levels of the same parity. The prescriptions of the continuous time quantum random walk are applied to the single molecule, interpreted as an open quantum system, and the master equation driving its internal dynamics is built for a general distribution of the waiting times between two consecutive collisions. The coherence terms and the populations of the energy levels relax to the asymptotics with inverse power laws for relevant classes of non-Poissonian distributions of the collision times. The stable asymptotic equilibrium configuration is independent of the distribution. The long time dynamics may be hindered by increasing the tail of the distribution density. This effect may be interpreted as the appearance of the quantum Zeno effect over long time scales.

quant-ph↗

arXiv2011

Anomalous decay of an atom in structured band gap reservoirs

We analyze the spontaneous emission of a two-level atom interacting with a special class of structured reservoirs of field modes with band gap edge coinciding with the atomic transition frequency. The exact time evolution of the population of the excited level is evaluated analytically through series of Fox-$H$ functions. Over estimated long time scales, inverse power law relaxations emerge, with powers decreasing continuously to 2 according to the choice of the special reservoir. No trapping of the population of the excited level emerges. The same results are recovered in presence of $N-1$ atoms, each one in the ground state, described by the Dicke model. The power of the inverse power law decay results to be independent of $N$. A critical number $N_α^{(\star)}$ is evaluated, such that, for $N \gg N_α^{(\star)}$, the inverse power law decay vanishes.

quant-ph↗