arXiv subjects

D. Puertas-Centeno

Publications and source records attributed to D. Puertas-Centeno.

14 recordsLinked to original sources

arXiv2025

Sundman-like transformations and the NRT nonlinear Schrödinger equation

We present a new generalization of the well-known power-type Sundman transformation, involving not only powers of the function but also of its derivative, along with its inverse. Our aim is to explore the use of such transformations in the derivation of solutions of ordinary differential equations and in the study of their properties. We then show their usefulness in the framework of the nonlinear Nobre--Reigo-Monteiro--Tsallis (NRT) nonlinear Schrödinger equation. More precisely, we employ them to analyze a family of ordinary differential equations which includes the Lorentzian solutions of the NRT-nonlinear Schrödinger equation for a constant potential. Moreover, an explicit expression for the Lorentzian solitary wave solutions is given, for any real value of the non-linearity parameter q, in terms of a transformation depending on q applied to the classical Lorentzian solution with q = 1, i.e., we succeed in encapsulating the whole nonlinear behavior in the new transformations. In addition, the composition of this transformation with its inverse (with different parameters) allows us to perform a shift in the nonlinearity parameter q. Moreover, a certain subfamily of our generalized transformations, which perform a shift on the non-linearity parameter q of the Lorentzian solutions, is found to have a group structure. The same subfamily of transformations allows us, again, to perform a shift in the non-linearity parameter q, but in this case in the traveling wave solution for a free particle.

nlin.SI↗

arXiv2025

Generalized and new solutions of the NRT nonlinear Schrödinger equation

In this paper we present new solutions of the non-linear Schröodinger equation proposed by Nobre, Rego-Monteiro and Tsallis for the free particle, obtained from different Lie symmetry reductions. Analytical expressions for the wave function, the auxiliary field and the probability density are derived using a variety of approaches. Solutions involving elliptic functions, Bessel and modified Bessel functions, as well as the inverse error function are found, amongst others. On the other hand, a closed-form expression for the general solution of the traveling wave ansatz (see Bountis and Nobre) is obtained for any real value of the nonlinearity index. This is achieved through the use of the so-called generalized trigonometric functions as defined by Lindqvist and Drábek, the utility of which in analyzing the equation under study is highlighted throughout the paper.

math-ph↗

arXiv2021

Multidimensional hydrogenic states: Position and momentum expectation values

The position and momentum probability densities of a multidimensional quantum system are fully characterized by means of the radial expectation values $\langle r^α\rangle$ and $\left\langle p^α\right\rangle$, respectively. These quantities, which describe and/or are closely related to various fundamental properties of realistic systems, have not been calculated in an analytical and effective manner up until now except for a number of three-dimensional hydrogenic states. In this work we explicitly show these expectation values for all discrete stationary $D$-dimensional hydrogenic states in terms of the dimensionality $D$, the strength of the Coulomb potential (i.e., the nuclear charge) and the $D$ state's hyperquantum numbers. Emphasis is placed on the momentum expectation values (mostly unknown, specially the ones with odd order) which are obtained in a closed compact form. Applications are made to circular, $S$-wave, high-energy (Rydberg) and high-dimensional (pseudo-classical) states of three- and multidimensional hydrogenic atoms. This has been possible because of the analytical algebraic and asymptotical properties of the special functions (orthogonal polynomials, hyperspherical harmonics) which control the states' wavefunctions. Finally, some Heisenberg-like uncertainty inequalities satisfied by these dispersion quantities are also given and discussed.

quant-ph↗

arXiv2020

Crámer-Rao complexity of the two-dimensional confined hydrogen

The internal disorder of the two-dimensional confined hydrogenic atom is numerically studied in terms of the confinement radius for the 1\textit{s}, 2\textit{s}, 2\textit{p} and 3\textit{d} quantum states by means of the statistical Crámer-Rao complexity measure. First, the confinement dependence of the variance and the Fisher information of the position and momentum spreading of its electron distribution are computed and discussed. Then, the Crámer-Rao complexity measure (which quantifies the combined balance of the charge concentration around the mean value and the gradient content of the electron distribution) is investigated in position and momentum spaces. We found that confinement does distinguish complexity of the system for all quantum states by means of this two component measure.

quant-ph↗

arXiv2020

Two-dimensional confined hydrogen: An entropy and complexity approach

The position and momentum spreading of the electron distribution of the two-dimensional confined hydrogenic atom, which is a basic prototype of the general multidimensional confined quantum systems, is numerically studied in terms of the confinement radius for the 1s, 2s, 2p and 3d quantum states by means of the main entropy and complexity information-theoretic measures. First, the Shannon entropy and the Fisher information as well as the associated uncertainty relations are computed and discussed. Then, the Fisher-Shannon, LMC and LMC-Rényi complexity measures are examined and mutually compared. We have found that these entropy and complexity quantities reflect the rich properties of the electron confinement extent in the two conjugated spaces.

quant-ph↗

arXiv2018

Differential-escort transformations and the monotonicity of the LMC-Rényi complexity measure

Escort distributions have been shown to be very useful in a great variety of fields ranging from information theory, nonextensive statistical mechanics till coding theory, chaos and multifractals. In this work we give the notion and the properties of a novel type of escort density, the differential-escort densities, which have various advantages with respect to the standard ones. We highlight the behavior of the differential Shannon, Rényi and Tsallis entropies of these distributions. Then, we illustrate their utility to prove the monotonicity property of the LMC-Rényi complexity measure and to study the behavior of general distributions in the two extreme cases of minimal and very high LMC-Rényi complexity. Finally, this transformation allows us to obtain the Tsallis q-exponential densities as the differential-escort transformation of the exponential density.

math-ph↗

arXiv2018

Rényi entropies for multidimensional hydrogenic systems in position and momentum spaces

The Rényi entropies of Coulomb systems $R_{p}[ρ], 0 < p < \infty$ are logarithms of power functionals of the electron density $ρ(\vec{r})$ which quantify most appropriately the electron uncertainty and describe numerous physical observables. However, its analytical determination is a hard issue not yet solved except for the first lowest-lying energetic states of some specific systems. This is so even for the $D$-dimensional hydrogenic system, which is the main prototype of the multidimensional Coulomb many-body systems. Recently, the Rényi entropies of this system have been found in the two extreme high-energy (Rydberg) and high-dimensional (pseudo-classical) cases. In this work we determine the position and momentum Rényi entropies (with integer $p$ greater than 1) for all the discrete stationary states of the multidimensional hydrogenic system directly in terms of the hyperquantum numbers which characterize the states, the nuclear charge and the space dimensionality. We have used a methodology based on linearization formulas for powers of the orthogonal Laguerre and Gegenbauer polynomials which control the hydrogenic states.

quant-ph↗

arXiv2018

Exact Rényi entropies of $D$-dimensional harmonic systems

The determination of the uncertainty measures of multidimensional quantum systems is a relevant issue \textit{per se} and because these measures, which are functionals of the single-particle probability density of the systems, describe numerous fundamental and experimentally accessible physical quantities. However, it is a formidable task (not yet solved, except possibly for the ground and a few lowest-lying energetic states) even for the small bunch of elementary quantum potentials which are used to approximate the mean-field potential of the physical systems. Recently, the dominant term of the Heisenberg and Rényi measures of the multidimensional harmonic system (i.e., a particle moving under the action of a $D$-dimensional quadratic potential, $D > 1$) has been analytically calculated in the high-energy (i.e., Rydberg) and the high-dimensional (i.e., pseudoclassical) limits. In this work we determine the exact values of the Rényi uncertainty measures of the $D$-dimensional harmonic system for all ground and excited quantum states directly in terms of $D$, the potential strength and the hyperquantum numbers.

math-ph↗

arXiv2017

Entropic uncertainty measures for large dimensional hydrogenic systems

The entropic moments of the probability density of a quantum system in position and momentum spaces describe not only some fundamental and/or experimentally accessible quantities of the system, but also the entropic uncertainty measures of Rényi type which allow one to find the most relevant mathematical formalizations of the position-momentum Heisenberg's uncertainty principle, the entropic uncertainty relations. It is known that the solution of difficult three-dimensional problems can be very well approximated by a series development in $1/D$ in similar systems with a non-standard dimensionality $D$; moreover, several physical quantities of numerous atomic and molecular systems have been numerically shown to have values in the large-$D$ limit comparable to the corresponding ones provided by the three-dimensional numerical self-consistent field methods. The $D$-dimensional hydrogenic atom is the main prototype of the physics of multidimensional many-electron systems. In this work we rigorously determine the leading term of the Rényi entropies of the $D$-dimensional hydrogenic atom at the limit of large $D$. As a byproduct, we show that our results saturate the known position-momentum Rényi-entropy-based uncertainty relations.

quant-ph↗

arXiv2017

Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

In this work we find that not only the Heisenberg-like uncertainty products and the Rényi-entropy-based uncertainty sum have the same first-order values for all the quantum states of the $D$-dimensional hydrogenic and oscillator-like systems, respectively, in the pseudoclassical ($D \to \infty$) limit but a similar phenomenon also happens for both the Fisher-information-based uncertainty product and the Shannon-entropy-based uncertainty sum, as well as for the Crámer-Rao and Fisher-Shannon complexities. Moreover, we show that the LMC (López-Ruiz-Mancini-Calvet) and LMC-Rényi complexity measures capture the hydrogenic-harmonic difference in the high dimensional limit already at first order.

quant-ph↗

arXiv2017

The biparametric Fisher-Rényi complexity measure and its application to the multidimensional blackbody radiation

We introduce a biparametric Fisher-Rényi complexity measure for general probability distributions and we discuss its properties. This notion, which is composed of two entropy-like components (the Rényi entropy and the biparametric Fisher information), generalizes the basic Fisher-Shannon measure and the previous complexity quantifiers of Fisher-Rényi type. Then, we illustrate the usefulness of this notion by carrying out a information-theoretical analysis of the spectral energy density of a $d$-dimensional blackbody at temperature $T$. It is shown that the biparametric Fisher-Rényi measure of this quantum system has a universal character in the sense that it does not depend on temperature nor on any physical constant (e.g., Planck constant, speed of light, Boltzmann constant), but only on the space dimensionality $d$. Moreover, it decreases when $d$ is increasing, but exhibits a non trivial behavior for a fixed $d$ and a varying parameter, which somehow brings up a non standard structure of the blackbody $d$-dimensional density distribution.

quant-ph↗

arXiv2017

Heisenberg and entropic uncertainty measures for large-dimensional harmonic systems

The $D$-dimensional harmonic system (i.e., a particle moving under the action of a quadratic potential) is, together with the hydrogenic system, the main prototype of the physics of multidimensional quantum systems. In this work we rigorously determine the leading term of the Heisenberg-like and entropy-like uncertainty measures of this system as given by the radial expectation values and the Rényi entropies, respectively, at the limit of large $D$. The associated multidimensional position-momentum uncertainty relations are discussed, showing that they saturate the corresponding general ones. A conjecture about the Shannon-like uncertainty relation is given and an interesting phenomenon is observed: the Heisenberg-like and Rényi-entropy-based equality-type uncertainty relations for all the $D$-dimensional harmonic oscillator states in the pseudoclassical ($D \to \infty$) limit are the same as the corresponding ones for the hydrogenic systems, despite the so different character of the oscillator and Coulomb potentials.

quant-ph↗

arXiv2016

Entropic measures of Rydberg-like harmonic states

The Shannon entropy, the desequilibrium and their generalizations (Rényi and Tsallis entropies) of the three-dimensional single-particle systems in a spherically-symmetric potential $V(r)$ can be decomposed into angular and radial parts. The radial part depends on the analytical form of the potential, but the angular part does not. In this paper we first calculate the angular entropy of any central potential by means of two analytical procedures. Then, we explicitly find the dominant term of the radial entropy for the highly energetic (i.e., Rydberg) stationary states of the oscillator-like systems. The angular and radial contributions to these entropic measures are analytically expressed in terms of the quantum numbers which characterize the corresponding quantum states and, for the radial part, the oscillator strength. In the latter case we use some recent powerful results of the information theory of the Laguerre polynomials and spherical harmonics which control the oscillator-like wavefunctions.

quant-ph↗

arXiv2016

Entropic properties of $D$-dimensional Rydberg systems

The fundamental information-theoretic measures (the Rényi $R_{p}[ρ]$ and Tsallis $T_{p}[ρ]$ entropies, $p>0$) of the highly-excited (Rydberg) quantum states of the $D$-dimensional ($D>1$) hydrogenic systems, which include the Shannon entropy ($p \to 1$) and the disequilibrium ($p = 2$), are analytically determined by use of the strong asymptotics of the Laguerre orthogonal polynomials which control the wavefunctions of these states. We first realize that these quantities are derived from the entropic moments of the quantum-mechanical probability $ρ(\vec{r})$ densities associated to the Rydberg hydrogenic wavefunctions $Ψ_{n,l,\{μ\}}(\vec{r})$, which are closely connected to the $\mathfrak{L}_{p}$-norms of the associated Laguerre polynomials. Then, we determine the ($n\to\infty$)-asymptotics of these norms in terms of the basic parameters of our system (the dimensionality $D$, the nuclear charge and the hyperquantum numbers $(n,l,\{μ\}$) of the state) by use of recent techniques of approximation theory. Finally, these three entropic quantities are analytically and numerically discussed in terms of the basic parameters of the system for various particular states.

quant-ph↗