M. Gadella
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The One‐Dimensional Coulomb Hamiltonian: Properties of Its Birman–Schwinger Operator Open
The objective of the present paper is to study in detail the properties of the Birman–Schwinger operator for a self‐adjoint realization of the one‐dimensional Hamiltonian with the Coulomb potential, both when the Hamiltonian is defined onl…
An Extended Picard Method to solve non-linear systems of ODE with applications Open
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 me…
Some point and singular potentials in one dimension: Standard arguments and new advances.<sup>1</sup> Open
We review some methods to obtain self-adjoint realizations of one dimensional Hamiltonians with contact interactions, such as Dirac deltas or their first derivatives, or singular interactions, such us the one dimensional Coulomb interactio…
RHS and Quantum Mechanics: Some Extra Examples Open
Rigged Hilbert spaces (RHSs) are the right mathematical context that include many tools used in quantum physics, or even in some chaotic classical systems. It is particularly interesting that in RHS, discrete and continuous bases, as well …
Gelfand Triplets, Ladder Operators and Coherent States Open
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 th…
RHS and Quantum Mechanics: Some Extra Examples Open
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 bas…
A note on linear differential equations with variable coefficients Open
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 ar…
Gelfand Triplets, Ladder Operators and Coherent States Open
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…
The one-dimensional Coulomb Hamiltonian: Properties of its Birman-Schwinger operator Open
We study the Birman-Schwinger operator for a self-adjoint realisation of the one-dimensional Hamiltonian with the Coulomb potential. We study both the case in which this Hamiltonian is defined on the whole real line and when it is only def…
Analysis of a one-dimensional Hamiltonian with a singular double well consisting of two nonlocal $$\delta '$$ interactions Open
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 $$\delta '$$ -interactions of equal strength and symmetrically located with respect …
An Extended Picard Method to solve non-linear systems of ODE. Some applications to chemical reactions Open
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 me…
Gelfand Triplets, Continuous and Discrete Bases and Legendre Polynomials Open
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 o…
Generalized Heisenberg-Weyl groups and Hermite functions Open
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 subgrou…
SUSY partners and S-matrix poles of the one-dimensional Rosen–Morse II potential Open
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 …
Analysis of a one-dimensional Hamiltonian with a singular double well consisting of two nonlocal $δ'$ interactions Open
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…
SUSY partners and $S$-matrix poles of the one dimensional Rosen-Morse II Hamiltonian Open
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…
A variational modification of the Harmonic Balance method to obtain approximate Floquet exponents Open
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 th…
Report on scipost_202212_00018v1 Open
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 subgroup…
An Algebraic Model for Quantum Unstable States Open
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 d…
A modification of a classical method to obtain Floquet exponents and solutions for linear periodic differential equations. Open
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 th…
On Hermite Functions, Integral Kernels, and Quantum Wires Open
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 scal…
A modified Lyapunov method and its applications to ODE Open
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 approxim…
Averages of observables on Gamow states Open
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 th…
Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian: Special Cases Open
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 selfad…
Mathematical Models for Unstable Quantum Systems and Gamow States Open
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 const…
Symmetry Groups, Quantum Mechanics and Generalized Hermite Functions Open
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 famil…
A modified Lyapunov method and its applications to ODE Open
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 approxi…