Manuel Gadella

University of Valladolid | UVA · Department of Theoretical Physics, Atomic and Optics

Inspired by a similar construction on Hermite functions, we construct two series of Gelfand triplets, each one spanned by Laguerre–Gauss functions with a fixed positive value of one parameter, considered as the fundamental one. We prove the continuity of different types of ladder operators on these triplets. Laguerre–Gauss functions with negative v...

The rigged Hilbert spaces (RHS) are the right mathematical context which includes many tools used in quantum physics, or even in some chaotic classical systems. It is particularly interesting that in RHS coexist discrete and continuous basis, abstract basis along basis of special functions and representations of Lie algebras of symmetries by contin...

We consider a basis of square integrable functions on a rectangle, contained in R2, 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 interva...

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 gi...

In the present paper and inspired with a similar construction on Hermite functions, we construct two series of Gelfand triplets each one spanned by Laguerre-Gauss functions with a fixed positive value of one of their parameters, considered as the fundamental one. We prove the continuity of different types of ladder operators on these triplets. Lagu...

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 renormalisation of the coupling constant in order to rigorously achieve a self-ad...

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 renormalisation of the coupling constant in order to rigorously achieve a self-ad...

A generalisation of Euclidean and pseudo-Euclidean groups is presented, where the Weyl-Heisenberg groups, well known in quantum mechanics, are involved. A new family of groups is obtained including all the above-mentioned groups as subgroups. Symmetries, like self-similarity and invariance with respect to the orientation of the axes, are properly i...

Among the list of one-dimensional solvable Hamiltonians, we find the Hamiltonian with the Rosen–Morse II potential. The first objective is to analyse 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...

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...

We propose a modification of a method based on Fourier analysis to obtain the Floquet characteristic exponents for periodic homogeneous linear systems, which shows a high precision. This modification uses a variational principle to find the correct Floquet exponents among the solutions of an algebraic equation. Once we have these Floquet exponents,...

In this review, we present a rigorous construction of an algebraic method for quantum unstable states, also called Gamow states. A traditional picture associates these states to vectors states called Gamow vectors. However, this has some difficulties. In particular, there is no consistent definition of mean values of observables on Gamow vectors. I...

We propose a modification of a method based on Fourier analysis to obtain the Floquet characteristic exponents for periodic homogeneous linear systems, which shows a high precision. This modification uses a variational principle to find the correct Floquet exponents among the solutions of an algebraic equation. Once we have these Floquet exponents,...

In this note, we first evaluate and subsequently achieve a rather accurate approximation of a scalar product, the calculation of which is essential in order to determine the ground state energy in a two-dimensional quantum model. This scalar product involves an integral operator defined in terms of the eigenfunctions of the harmonic oscillator, exp...

Here, we propose a method to obtain local analytic approximate solutions of ordinary differential equations with variable coefficients, or even some nonlinear equations, inspired in the Lyapunov method, where instead of polynomial approximations, we use truncated Fourier series with variable coefficients as approximate solutions. In the case of equ...

In this note we first evaluate and subsequently achieve a rather accurate approximation of a scalar product, the calculation of which is essential in order to determine the ground state energy in a two-dimensional quantum model. This scalar product involves an integral operator defined in terms of the eigenfunctions of the harmonic oscillator, expr...

We propose a formulation of Gamow states, which is the part of unstable quantum states that decays exponentially, with two advantages in relation with the usual formulation of the same concept using Gamow vectors. The first advantage is that this formulation shows that Gamow states cannot be pure states, so that they may have a non-zero entropy. Th...

In a previous paper, we used a classification of the self adjoint extensions, also called self-adjoint determinations, of the differential operator −d2/dx2 in order to obtain the whole list of Supersymmetric (SUSY) partners of those selfadjoint determinations for which the ground state has strictly positive energy. The existence of self adjoint det...

We review some results in the theory of non-relativistic quantum unstable systems. We account for the most important definitions of quantum resonances that we identify with unstable quantum systems. Then, we recall the properties and construction of Gamow states as vectors in some extensions of Hilbert spaces, called Rigged Hilbert Spaces. Gamow st...

This is a review paper on the generalization of Euclidean as well as pseudo-Euclidean groups of interest in quantum mechanics. The Weyl–Heisenberg groups, Hn, together with the Euclidean, En, and pseudo-Euclidean Ep,q, groups are two families of groups with a particular interest due to their applications in quantum physics. In the present manuscrip...

Here, we propose a method to obtain local analytic approximate solutions of ordinary differential equations with variable coefficients, or even some non-linear equations, inspired in the Lyapunov method, where instead of polynomial approximations, we use truncated Fourier series with variable coefficients as approximate solutions. In the case of eq...

In this article, we provide an expansion (up to the fourth order of the coupling constant) of the energy of the ground state of the Hamiltonian of a quantum mechanical particle moving inside a parabolic well in the x-direction and constrained by the presence of a two-dimensional impurity, modelled by an attractive two-dimensional isotropic Gaussian...

We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases...

In this paper, we provide a detailed description of the eigenvalue \( E_{D}(x_0)\le 0\) (respectively, \( E_{N}(x_0)\le 0\)) of the self-adjoint Hamiltonian operator representing the negative Laplacian on the positive half-line with a Dirichlet (resp. Neuman) boundary condition at the origin perturbed by an attractive Dirac distribution \(-\lambda...

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(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 met...

We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator $-d^2/dx^2$ on $L^2[-a,a]$, $a>0$, that is, the one dimensional infinite square well. First of all, we classify these self-adjoint extensions in terms of several choices of the parameters determining each of the exten...

We develop a one‐step matrix method in order to obtain approximate solutions of first‐order nonlinear systems and nonlinear ordinary differential equations, reducible to first‐order systems. We find a sequence of such solutions that converge to the exact solution. We apply the method to different well‐known examples and check its precision, in term...

In this paper we provide a detailed description of the eigenvalue $ E_{D}(x_0)\leq 0$ (respectively $ E_{N}(x_0)\leq 0$) of the self-adjoint Hamiltonian operator representing the negative Laplacian on the positive half-line with a Dirichlet (resp. Neuman) boundary condition at the origin perturbed by an attractive Dirac distribution $-\lambda \delt...

We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator −d2/dx2 on L2[−a,a], a>0, that is, the one dimensional infinite square well. First of all, we classify these self-adjoint extensions in terms of several choices of the parameters determining each of the extensions. Th...

We present a detailed study of a generalised one-dimensional Kronig-Penney model using δ-δ potentials. We analyse the band structure and the density of states in two situations. In the first case, we consider an infinite array formed by identical δ-δ potentials standing at the linear lattice nodes. This case will be known throughout the paper as th...

We develop a one step matrix method in order to obtain approximate solutions of first order systems and non-linear ordinary differential equations, reducible to first order systems. We find a sequence of such solutions that converge to the exact solution. We study the precision, in terms of the local error, of the method by applying it to different...

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 (c...

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. I...

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...

Contains the basic notions which support the use of Statistical Mechanics for unstable systems. The chapter is devoted to the discussion of an issue that has rarely been treated in the Literature: the possibility of assigning thermodynamic variables to unstable quantum systems. The interest of this study lies in the fact that many of existing quant...

Explores the principles of statistical mechanics in terms of geometry, as anticipated by Bogolubov, and the role of dynamical variables in the statistical properties of a system, either classical or quantum. The evolution of these variables is described by a non-linear system of 2n differential equations.

This chapter contains the notions of operators and their role in statistical mechanics, particularly in the combined time-temperature representations. The chapter is devoted to the unification of these schemes, by paying attention to the definition of thermodynamical observables in the context of static and quasi-static processes. The line of discu...

The chapter focusses on the question concerning the relationships between the equations of motion in Lagrangian mechanics with the probabilistic interpretation of statistical mechanics. The motivation is obvious, since the probabilistic conception of statistical mechanics is based on the knowledge of the exact energy levels of the system under stud...

Specializes in the connections between statistical ensembles, from the point of view of the Hamiltonian dynamics and from the operator approach. The chapter is devoted to the unification of both of these schemes, paying attention to the definition of thermodynamical observables in the context of static and quasi-static processes.

This chapter contains a revision of concepts and methods in classical and quantum statistical mechanics, for discrete and continuous spectra, based on the notion of probabilities and thermal equilibrium in the Gibbs approach. This formulation applies to both classical and quantum systems. The concept of probability is illustrated by solving a syste...

Reviews the Feynman path integral formulation of propagators and the connections with statistical mechanics at the level of partition functions. This chapter is devoted to the unification of both schemes, the one based on the notion of probabilities and Feynman approach based on amplitudes.

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 by means of the Fourier transform and the discrete Fourier transform. This relation is unitary. Although these bases are not orthonormal the Gramm-Schmidt method permits...

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...

We introduced some contact potentials that can be written as a linear combination of the Dirac delta and its first derivative, the $\delta$-$\delta'$ interaction. After a simple general presentation in one dimension, we briefly discuss a one dimensional periodic potential with a $\delta$-$\delta'$ interaction at each node. The dependence of energy...

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 imp...

We analyze the structure of the scattering matrix, S(k), for the one dimensional Morse potential. We show that, in addition to a finite number of bound state poles and an infinite number of anti-bound poles, there exist an infinite number of redundant poles, on the positive imaginary axis, which do not correspond to either of the other types. This...

We analyze the structure of the scattering matrix, $S(k)$, for the one dimensional Morse potential. We show that, in addition to a finite number of bound state poles and an infinite number of anti-bound poles, there exist an infinite number of redundant poles, on the positive imaginary axis, which do not correspond to either of the other types. Thi...

We study the time-dependent Schrödinger equation with finite number of Dirac δ and δ′ potentials with time dependent strengths in one dimension. We obtain the formal solution for generic time dependent strengths and then we study the particular cases for single delta potential and limiting cases for finitely many delta potentials. Finally, we inves...

We study a non-relativistic particle subject to a three-dimensional spherical potential consisting of a finite well and a radial \(\delta -\delta '\) contact interaction at the well edge. This contact potential is defined by appropriate matching conditions for the radial functions, thereby fixing a self-adjoint extension of the non-singular Hamilto...

The fractional calculus is useful to model nonlocal 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 nonlocality of the fractional derivative, we may esta...

This paper is a contribution to the study of the relations between special functions, Lie algebras, and rigged Hilbert spaces. The discrete indices and continuous variables of special functions are in correspondence with the representations of their algebra of symmetry, which induce discrete and continuous bases coexisting on a rigged Hilbert space...

This book presents a variety of techniques for tackling phenomena that are not amenable to the conventional approach based on the concept of probabilities. The methods described rely on the use of path integration, thermal Green functions, time-temperature propagators, Liouville operators, second quantization, and field correlators at finite densit...

We study a simplified mean-field potential consisting of a finite radial square well plus a radial $\delta$-$\delta'$ contact potential in order to describe single neutron energy levels. This is a limiting case of the Woods-Saxon potential, where the Dirac-$\delta$ takes account of the nuclear spin-orbit interaction. This contact potential is defin...

This paper is a contribution to the study of the relations between special functions, Lie algebras and rigged Hilbert spaces. The discrete indices and continuous variables of special functions are in correspondence with the representations of their algebra of symmetry, that induce discrete and continuous bases coexisting on a rigged Hilbert space s...

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 es...

We present a detailed study of a generalised one-dimensional Kronig-Penney model using $\delta\text{-}\delta'$ potentials. We analyse the band structure and the density of states in two situations. In the first case we consider an infinite array formed by identical $\delta\text{-}\delta'$ potentials standing at the linear lattice nodes. This case w...

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 as 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 areas of interest of Z...

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 modeled by an attractive Gaussian potential. We investigate the Birman-Schwinger operator associated to a model assuming the presence of a Gaussian impuri...

We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible represent...

We show how a proper use of the Lippmann–Schwinger equation simplifies the calculations to obtain scattering states for one dimensional systems perturbed by N Dirac delta equations. Here, we consider two situations. In the former, attractive Dirac deltas perturbed the free one dimensional Schrödinger Hamiltonian. We obtain explicit expressions for...

We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible represent...

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...

We study three solvable two-dimensional systems perturbed by a point interaction centred at the origin. The unperturbed systems are the isotropic harmonic oscillator, a square pyramidal potential and a combination thereof. We study the spectrum of the perturbed systems. We show that, while most eigenvalues are not affected by the point perturbation...

We have defined a pair of families of coherent states using creation and annihilation operators relating the Gamow states, vector states for non-relativistic quantum resonances. We have used an explicit one dimensional model, for which these operators may be explicitly written: the one dimensional Pöschl–Teller Hamiltonian. We have shown that this...

We revise the symmetries of the Zernike polynomials that determine the Lie algebra su(1,1) x 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 functi...

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 on the use of rigged Hilbert spaces. We find a relation betw...

We present a family of unitary irreducible representations of SU(2) realized in the plane, in terms of the associated Laguerre polynomials. These functions are similar to the spherical harmonics defined on the sphere. Relations with a space of square integrable functions defined on the plane, \(L^2({\mathbb R}^2)\), are analyzed. We have also enlar...

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 betwe...

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 no...

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 no...

In this brief presentation, some striking differences between level crossings of eigenvalues in one dimension (harmonic or conic oscillator with a central nonlocal δ-interaction) or three dimensions (isotropic harmonic oscillator with a three-dimensional delta located at the origin) and those occurring in the two-dimensional analogue of these model...

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...

We discuss a method based on a segmentary approximation of solutions of the Schr\"odinger 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 de...

The discrete spectrum of the Hamiltonian of the 2D isotropic harmonic oscillator and that of the Hamiltonian of the 2D isotropic pyramidal oscillator are thoroughly investigated with particular attention paid to level crossings of eigenvalues in both spectra.

This paper is devoted to study discrete and continuous bases for spaces supporting representations of SO(3) and SO(3,2) where the spherical harmonics are involved. We show how discrete and continuous bases coexist on appropriate choices of rigged Hilbert spaces. We prove the continuity of relevant operators and the operators in the algebras spanned...

In this paper, we propose a pedagogical presentation of the Lippmann-Schwinger equation as a powerful tool, so as to obtain important scattering information. In particular, we consider a one-dimensional system with a Schrödinger-type free Hamiltonian decorated with a sequence of N attractive Dirac delta interactions. We first write the Lippmann-Sch...

p>We propose a new approach to the problem of finding the eigenvalues (energy levels) in the discrete spectrum of the one-dimensional Hamiltonian with an attractive Gaussian potential by using the well-known Birman-Schwinger technique. However, in place of the Birman-Schwinger integral operator we consider an isospectral operator in momentum space,...

We consider the one-dimensional Hamiltonian with a V-shaped potential H0=[Formula presented]−[Formula presented]+|x|, decorated with a point impurity of either δ-type, or local δ′-type or even nonlocal δ′-type, thus yielding three exactly solvable models. We analyse the behaviour of the change in the energy levels when an interaction of the type −λ...

It is well known that related with the irreducible representations of the Lie group $SO(2)$ we find a discrete basis as well a continuous one. In this paper we revisited this situation under the light of Rigged Hilbert spaces, which are the suitable framework to deal with both discrete and bases in the same context and in relation with physical app...

In this brief presentation some striking differences between level crossings of eigenvalues in one dimension (harmonic oscillator with a central delta'-interaction) or three dimensions (harmonic oscillator with a three-dimensional delta) and those occurring in their two-dimensional analogue will be pointed out. Joint work with L.M. Nieto, M. Gadell...