arXiv subjects

Manuel Gadella

Publications and source records attributed to Manuel Gadella.

At least 19 recordsLinked to original sources

arXiv2026

Bound States and Resonance Analysis of One-Dimensional Relativistic Parity-Symmetric Two Point Interactions

We consider the one-dimensional Dirac equation with the most general relativistic contact interaction supported on two points symmetrically located with respect to the origin. In order to determine the shape of the interaction, we use a distributional method, which in the present case is equivalent to the standard method of defining contact interactions by self-adjoint extensions of symmetric operators. The interaction on each of these two points depends on four parameters, each one having a clear physical meaning. We are interested in the scattering and confining properties of this model. We focus our attention on even or odd interactions under parity transformations and investigate the existence of critical and supercritical states, bound states, confinement and scattering resonances for some particular interactions of special interest.

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arXiv2025

An Extended Picard Method to solve non-linear systems of ODE with applications

We provide of a method to integrate first order non-linear systems of differential equations with variable coefficients. It determines approximate solutions given initial or boundary conditions or even for Sturm-Liouville problems. This method is a mixture between an iterative process, a la Picard, plus a segmentary integration, which gives explicit approximate solutions in terms of trigonometric functions and polynomials. The segmentary part is particularly important if the integration interval is large. This procedure provide a new tool so as to obtain approximate solutions of systems of interest in the analysis of chemical reactions. We test the method on some classical equations like Mathieu, Duffing quintic equation or Bratu's equation and have applied it on some models of chemical reactions.

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arXiv2024

A note on linear differential equations with variable coefficients

In this manuscript, we deal with some particular type of homogeneous first order linear systems with variable coefficients, in which we provide qualitative properties of the solution. When the coefficients of the indeterminate functions are periodic with the same period, $T$, we obtain a simple method so as to obtain the Floquet coefficients. We give a new interpretation of the averages on the interval $(0,T)$ of the matrix of the coefficients. We discuss some examples and made some comparison with previous results.

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arXiv2024

Analysis of a one-dimensional Hamiltonian with a singular double well consisting of two nonlocal $δ'$ interactions

The objective of the present paper is the study of a one-dimensional Hamiltonian with the interaction term given by the sum of two nonlocal attractive $δ'$-interactions of equal strength and symmetrically located with respect to the origin. We use the procedure known as {\it renormalisation of the coupling constant} in order to rigorously achieve a self-adjoint determination for this Hamiltonian. This model depends on two parameters, the interaction strength and the distance between the centre of each interaction and the origin. Once we have the self-adjoint determination, we obtain its discrete spectrum showing that it consists of two negative eigenvalues representing the energy levels. We analyse the dependence of these energy levels on the above-mentioned parameters. We investigate the possible resonances of the model. Furthermore, we analyse in detail the limit of our model as the distance between the supports of the two $δ'$ interactions vanishes.

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arXiv2023

Gelfand Triplets, Continuous and Discrete Bases and Legendre Polynomials

We consider a basis of square integrable functions on a rectangle, contained in $R^2$, constructed with Legendre polynomials, suitable, for instance, for the analogical description of images on the plane or in other fields of application of the Legendre polynomials in higher dimensions. After extending the Legendre polynomials to any arbitrary interval of the form [a,b], from its original form on [-1,1], we generalize the basis of Legendre polynomials to two dimensions. This is the first step to generalize the basis to n-dimensions. We present some mathematical constructions such as Gelfand triples appropriate on this context. ``Smoothness'' of functions on space of test functions and some other properties are revisited, as well as te continuity of generators of $su(1,1)$ on this context.

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arXiv2023

SUSY partners and $S$-matrix poles of the one dimensional Rosen-Morse II Hamiltonian

Among the list of one dimensional solvable Hamiltonians, we find the Hamiltonian with the Rosen--Morse II potential. The first objective is to analyze the scattering matrix corresponding to this potential. We show that it includes a series of poles corresponding to the types of redundant poles or anti-bound poles. In some cases, there are even bound states and this depends on the values of given parameters. Then, we perform different supersymmetric transformations on the original Hamiltonian either using the ground state (for those situations where there are bound states) wave functions, or other solutions that come from anti-bound states or redundant states. We study the properties of these transformations.

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arXiv2021

Hermite functions and Fourier series

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle ($L^2(\mathcal C)$) and in $l_2(\mathbb Z)$, which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm--Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both $L^2(\mathcal C)$ and $l_2(\mathbb Z)$, so that all the mentioned operators are continuous.

math-ph↗

arXiv2020

Gamow vectors formalism applied to the Loschmidt echo

Gamow vectors have been developed in order to give a mathematical description for quantum decay phenomena. Mainly, they have been applied to radioactive phenomena, scattering and to some decoherence models. They play a crucial role in the description of quantum irreversible processes, and in the formulation of time asymmetry in quantum mechanics. In this paper, we use this formalism to describe a well-known phenomenon of irreversibility: the Loschmidt echo. The standard approach considers that the irreversibility of this phenomenon is the result of an additional term in the backward Hamiltonian. Here, we use the non-Hermitian formalism, where the time evolution is non-unitary. Additionally, we compare the characteristic decay times of this phenomenon with the decoherence ones. We conclude that the Loschmidt echo and the decoherence can be considered as two aspects of the same phenomenon, and that there is a mathematical relationship between their corresponding characteristic times.

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arXiv2020

Some recent results on contact or point supported potentials

We introduced some contact potentials that can be written as a linear combination of the Dirac delta and its first derivative, the $δ$-$δ'$ interaction. After a simple general presentation in one dimension, we briefly discuss a one dimensional periodic potential with a $δ$-$δ'$ interaction at each node. The dependence of energy bands with the parameters (coefficients of the deltas) can be computed numerically. We also study the $δ$-$δ'$ interaction supported on spheres of arbitrary dimension. The spherical symmetry of this model allows us to obtain rigorous conclusions concerning the number of bound states in terms of the parameters and the dimension. Finally, a $δ$-$δ'$ interaction is used to approximate a potential of wide use in nuclear physics, and estimate the total number of bound states as well as the behaviour of some resonance poles with the lowest energy.

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arXiv2020

The Birman-Schwinger Operator for a Parabolic Quantum Well in a Zero-Thickness Layer in the Presence of a Two-Dimensional Attractive Gaussian Impurity

In this note we consider a quantum mechanical particle moving inside an infinitesimally thin layer constrained by a parabolic well in the $x$-direction and, moreover, in the presence of an impurity modelled by an attractive Gaussian potential. We investigate the Birman-Schwinger operator associated to a model assuming the presence of a Gaussian impurity inside the layer and prove that such an integral operator is Hilbert-Schmidt, which allows the use of the modified Fredholm determinant in order to compute the bound states created by the impurity. Furthermore, we consider the case where the Gaussian potential degenerates to a $δ$-potential in the $x$-direction and a Gaussian potential in the $y$-direction. We construct the corresponding self-adjoint Hamiltonian and prove that it is the limit in the norm resolvent sense of a sequence of corresponding Hamiltonians with suitably scaled Gaussian potentials. Satisfactory bounds on the ground state energies of all Hamiltonians involved are exhibited.

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arXiv2019

Approximate solutions of one dimensional systems with fractional derivative

The fractional calculus is useful to model non-local phenomena. We construct a method to evaluate the fractional Caputo derivative by means of a simple explicit quadratic segmentary interpolation. This method yields to numerical resolution of ordinary fractional differential equations. Due to the non-locality of the fractional derivative, we may establish an equivalence between fractional oscillators and ordinary oscillators with a dissipative term.

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arXiv2019

Evolution of quantum observables: from non-commutativity to commutativity

A fundamental aspect of the quantum-to-classical limit is the transition from a non-commutative algebra of observables to commutative one. However, this transition is not possible if we only consider unitary evolutions. One way to describe this transition is to consider the Gamow vectors, which introduce exponential decays in the evolution. In this paper, we give two mathematical models in which this transition happens in the infite time limit. In the first one, we consider operators acting on the space of the Gamow vectors, which represent quantum resonances. In the second one, we use an algebraic formalism from scattering theory. We construct a non-commuting algebra which commutes in the infinite time limit.

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arXiv2019

Zernike functions, rigged Hilbert spaces and potential applications

We revise the symmetries of the Zernike polynomials that determine the Lie algebra su(1,1) + su(1,1). We show how they induce discrete as well continuous bases that coexist in the framework of rigged Hilbert spaces. We also discuss some other interesting properties of Zernike polynomials and Zernike functions. One of the interests of Zernike functions has been their applications in optics. Here, we suggest that operators on the spaces of Zernike functions may play a role in optical image processing.

math-ph↗

arXiv2018

Hermite functions, Lie groups and Fourier analysis

In this paper, we present recent results in harmonic analysis in the real line R and in the half-line R^+, which show a closed relation between Hermite and Laguerre functions, respectively, their symmetry groups and Fourier analysis. This can be done in terms of a unified framework based in the use of rigged Hilbert spaces. We find a relation between the universal enveloping algebra of the symmetry groups with the fractional Fourier transform. The results obtained are relevant in quantum mechanics as well as in signal processing as Fourier analysis has a close relation with signal filters. Also, we introduce some new results concerning a discretized Fourier transform on the circle. We introduce new functions on the circle constructed with the use of Hermite functions with interesting properties under Fourier transformations.

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arXiv2018

Dynamics of algebras in quantum unstable systems

We introduce a dynamical evolution operator for dealing with unstable physical process, such as scattering resonances, photon emission, decoherence and particle decay. With that aim, we use the formalism of rigged Hilbert space and represent the time evolution of quantum observables in the Heisenberg picture, in such a way that time evolution is non-unitary. This allows to describe observables that are initially non-commutative, but become commutative after time evolution. In other words, a non-abelian algebra of relevant observables becomes abelian when times goes to infinity. We finally present some relevant examples.

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arXiv2018

SU(2), Associated Laguerre Polynomials and Rigged Hilbert Spaces

We present a family of unitary irreducible representations of SU(2) realized in the plane, in terms of the Laguerre polynomials. These functions are similar to the spherical harmonics defined on the sphere. Relations with an space of square integrable functions defined on the plane, $L^2(\R^2)$, are analyzed. We have also enlarged this study using rigged Hilbert spaces that allow to work with iscrete and continuous bases like is the case here.

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arXiv2018

The Definition of Entropy for Quantum Unstable Systems: A View-Point Based on the Properties of Gamow States

In this paper, we review the concept of entropy in connection with the description of quantum unstable systems. We revise the conventional definition of entropy due to Boltzmann and extend it so as to include the presence of complex-energy states. After introducing a generalized basis of states which includes resonances, and working with amplitudes instead of probabilities, we~found an expression for the entropy which exhibits real and imaginary components. We discuss the meaning of the imaginary part of the entropy on the basis of the similarities existing between thermal and time evolutions.

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arXiv2018

A study of periodic potentials based on quadratic splines

We discuss a method based on a segmentary approximation of solutions of the Schrödinger by quadratic splines, for which the coefficients are determined by a variational method that does not require the resolution of complicated algebraic equations. The idea is the application of the method to one dimensional periodic potentials. We include the determination of the eigenvalues up to a given level, and therefore an approximation to the lowest energy bands. We apply the method to concrete examples with interest in physics and discussed the numerical errors.

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