arXiv subjects

M. A. Rego-Monteiro

Publications and source records attributed to M. A. Rego-Monteiro.

15 recordsLinked to original sources

arXiv2026

A Nonlinear $q$-Deformed Schrödinger Equation

We construct a new nonlinear deformed Schrödinger structure using a nonlinear derivative operator which depends on a parameter $q$. This operator recovers Newton derivative when $q \rightarrow 1$. Using this operator we propose a deformed Lagrangian which gives us a deformed nonlinear Schrödinger equation with a nonlinear kinetic energy term and a standard potential $V(\vec{x})$. We analytically solve the nonlinear deformed Schrödinger equation for $V(\vec{x}) = 0$ and $q \simeq1$. This model has a continuity equation, the energy is conserved, as well as the momentum and also interacts with electromagnetic field. Planck relation remains valid and in all steps we easily recover the undeformed quantities when the deformation parameter goes to 1. Finally, we numerically solve the equation of motion for the free particle in any spatial dimension, which shows a solitonic pattern when the space is equal to one for particular values of $q$.

nlin.PS↗

arXiv2024

Multi-parametric Nonlinear Generalization of Klein-Gordon: Real and Complex Fields

We construct a nonlinear multiparametric Klein-Gordon for complex and real fields with mass dimension depending on a real parameter $α$ as $δ= 2/(1+α)$ where $δ$ is the mass dimension of the fields. We show that there are three classes of generalized models, one class for complex fields and two different classes for real fields. All models in these three classes have travelling-wave solutions and satisfy the relativistic dispersion relation. Moreover, all models of the complex class and models of only one class of the two real classes recover the standard Klein-Gordon model. We also build the Lagrangian and the Hamiltonian for the three classes of models. The fields in the models of these three classes could in principle have the mass dimension varying from zero to one and this can allow us to construct interaction terms, other than $λΦ^4$, with coupling constants with positive or zero mass dimensions. Furthermore, we also show that there is a subclass of equations in the complex class which has a Lorentzian soliton solution.

quant-ph↗

arXiv2019

Lorentzian Solitary Wave in a Generalised Nonlinear Schrödinger Equation

We obtain a travelling-wave solution of a generalised nonlinear Schrödinger equation with an additional term of the form $Γ(ψ(x,t)) = λψ(x,t)^q$, where $λ$ and $q$ are real constants. Moreover, we show that the density of energy of the system for the travelling-wave solution presents a Lorentzian solitary wave behaviour.

nlin.PS↗

arXiv2016

Position-dependent mass quantum Hamiltonians: General approach and duality

We analyze a general family of position-dependent mass quantum Hamiltonians which are not self-adjoint and include, as particular cases, some Hamiltonians obtained in phenomenological approaches to condensed matter physics. We build a general family of self-adjoint Hamiltonians which are quantum mechanically equivalent to the non self-adjoint proposed ones. Inspired in the probability density of the problem, we construct an ansatz for the solutions of the family of self-adjoint Hamiltonians. We use this ansatz to map the solutions of the time independent Schrodinger equations generated by the non self-adjoint Hamiltonians into the Hilbert space of the solutions of the respective dual self-adjoint Hamiltonians. This mapping depends on both the position-dependent mass and on a function of position satisfying a condition that assures the existence of a consistent continuity equation. We identify the non self-adjoint Hamiltonians here studied to a very general family of Hamiltonians proposed in a seminal article of Harrison [1] to describe varying band structures in different types of metals. Therefore, we have self-adjoint Hamiltonians that correspond to the non self-adjoint ones found in Harrison's article. We analyze three typical cases by choosing a physical position-dependent mass and a deformed harmonic oscillator potential . We completely solve the Schrodinger equations for the three cases; we also find and compare their respective energy levels.

quant-ph↗

arXiv2008

Non-commutative reading of the complex plane through Delone sequences

The Berezin-Klauder-Toeplitz ("anti-Wick") quantization or "non-commutative reading" of the complex plane, viewed as the phase space of a particle moving on the line, is derived from the resolution of the unity provided by the standard (or gaussian) coherent states. The construction properties of these states and their attractive properties are essentially based on the energy spectrum of the harmonic oscillator, that is on the natural numbers. This work is an attempt for following the same path by considering sequences of non-negative numbers which are not "too far" from the natural numbers. In particular, we examine the consequences of such perturbations on the non-commutative reading of the complex plane in terms of its probabilistic, functional, and localization aspects.

quant-ph↗

arXiv2007

Generalized Heisenberg Algebras and Fibonacci Series

We have constructed a Heisenberg-type algebra generated by the Hamiltonian, the step operators and an auxiliar operator. This algebra describes quantum systems having eigenvalues of the Hamiltonian depending on the eigenvalues of the two previous levels. This happens, for example, for systems having the energy spectrum given by Fibonacci sequence. Moreover, the algebraic structure depends on two functions f(x) and g(x). When these two functions are linear we classify, analysing the stability of the fixed points of the functions, the possible representations for this algebra.

math-ph↗

arXiv2006

Low-Temperature Quasi-Equilibrium States in the Hydrogen Atom

The dynamics of the approach to equilibrium of the hydrogen atom is investigated numerically through a Monte Carlo procedure. We show that, before approaching ionization, the hydrogen atom may live in a quasi-equilibrium state, characterized by aging, whose duration increases exponentially as the temperatures decreases. By analyzing the quasi-equilibrium state, we compute averages of physical quantities for the hydrogen atom. We have introduced an analytic approach that fits satisfactorily the numerical estimates for low temperatures. Although the present analysis is expected to hold for energies typically up to 6% of the ionization energy, it works well for temperatures as high as 10^{4} K.

cond-mat.stat-mech↗

arXiv2004

Some boundary effects in quantum field theory

We have constructed a quantum field theory in a finite box, with periodic boundary conditions, using the hypothesis that particles living in a finite box are created and/or annihilated by the creation and/or annihilation operators, respectively, of a quantum harmonic oscillator on a circle. An expression for the effective coupling constant is obtained showing explicitly its dependence on the dimension of the box.

hep-th↗

arXiv2004

Construction of coherent states for physical algebraic systems

We construct a general state which is an eigenvector of the annihilation operator of the Generalized Heisenberg Algebra. We show for several systems, which are characterized by different energy spectra, that this general state satisfies the minimal set of conditions required to obtain Klauder's minimal coherent states.

math-ph↗

arXiv2002

Non-linear generalization of the sl(2) algebra

We present a generalization of the sl(2) algebra where the algebraic relations are constructed with the help of a general function of one of the generators. When this function is linear this algebra is a deformed sl(2) algebra. In the non-linear case, the finite dimensional representations are constructed in two different ways. In the first case, which provides finite dimensional representations only for the non-linear case, these representations come from solutions to a dynamical equation and we show how to construct explicitly these representations for a general quadratic non-linear function. The other type of finite dimensional representation comes from solutions to a cut condition equation. We give examples of solutions of this type in the non-linear case as well.

math-ph↗

arXiv2002

Generalized quantum field theory: perturbative computation and perspectives

We analyze some consequences of two possible interpretations of the action of the ladder operators emerging from generalized Heisenberg algebras in the framework of the second quantized formalism. Within the first interpretation we construct a quantum field theory that creates at any space-time point particles described by a q-deformed Heisenberg algebra and we compute the propagator and a specific first order scattering process. Concerning the second one, we draw attention to the possibility of constructing this theory where each state of a generalized Heisenberg algebra is interpreted as a particle with different mass.

hep-th↗

arXiv2001

Heisenberg-type structures of one-dimensional quantum Hamiltonians

We construct a Heisenberg-like algebra for the one dimensional infinite square-well potential in quantum mechanics. The ladder operators are realized in terms of physical operators of the system as in the harmonic oscillator algebra. These physical operators are obtained with the help of variables used in a recently developed non commutative differential calculus. This \textquotedblleft square-well algebra\textquotedblright is an example of an algebra in a large class of generalized Heisenberg algebras recently constructed. This class of algebras also contains $q$-oscillators as a particular case. We also discuss the physical content of this large class of algebras.

hep-th↗

arXiv2001

Construction of a non-standard quantum field theory through a generalized Heisenberg algebra

We construct a Heisenberg-like algebra for the one dimensional quantum free Klein-Gordon equation defined on the interval of the real line of length $L$. Using the realization of the ladder operators of this type Heisenberg algebra in terms of physical operators we build a 3+1 dimensional free quantum field theory based on this algebra. We introduce fields written in terms of the ladder operators of this type Heisenberg algebra and a free quantum Hamiltonian in terms of these fields. The mass spectrum of the physical excitations of this quantum field theory are given by $\sqrt{n^2 π^2/L^2+m_q^2}$, where $n= 1,2,...$ denotes the level of the particle with mass $m_q$ in an infinite square-well potential of width $L$.

hep-th↗

arXiv2001

Multi Parametric Deformed Heisenberg Algebras: A Route to Complexity

We introduce a generalization of the Heisenberg algebra which is written in terms of a functional of one generator of the algebra, $f(J_0)$, that can be any analytical function. When $f$ is linear with slope $θ$, we show that the algebra in this case corresponds to $q$-oscillators for $q^2 = \tan θ$. The case where $f$ is a polynomial of order $n$ in $J_0$ corresponds to a $n$-parameter deformed Heisenberg algebra. The representations of the algebra, when $f$ is any analytical function, are shown to be obtained through the study of the stability of the fixed points of $f$ and their composed functions. The case when $f$ is a quadratic polynomial in $J_0$, the simplest non-linear scheme which is able to create chaotic behavior, is analyzed in detail and special regions in the parameter space give representations that cannot be continuously deformed to representations of Heisenberg algebra.

hep-th↗

arXiv2000

Thermodynamic Properties of a Solid Exhibiting the Energy Spectrum given by the Logistic Map

We show that the infinite-dimensional representation of the recently introduced Logistic algebra can be interpreted as a non-trivial generalization of the Heisenberg or oscillator algebra. This allow us to construct a quantum Hamiltonian having the energy spectrum given by the logistic map. We analyze the Hamiltonian of a solid whose collective modes of vibration are described by this generalized oscillator and compute the thermodynamic properties of the model in the two-cycle and r=3.6785 chaotic region of the logistic map.

cond-mat.stat-mech↗