arXiv subjects

David J. Fernández C

Publications and source records attributed to David J. Fernández C.

13 recordsLinked to original sources

arXiv2023

Design of quasiperiodic magnetic superlattices and domain walls supporting bound states

We study the simplest Lamé magnetic superlattice in graphene, finding its allowed and forbidden energy bands and band-edge states explicitly. Then, we design quasiperiodic magnetic superlattices supporting bound states using Darboux transformations. This technique enables us to add any finite number of bound states, which we exemplify with the most straightforward cases of one and two bound states in the designed spectrum. The topics of magnetic superlattices and domain walls in gapped graphene turn out to be connected by a unitary transformation in the limit of significantly large oscillation periods. We show that the generated quasiperiodic magnetic superlattices are also linked to domain walls, with the bound states keeping their nature in such a limit.

cond-mat.mes-hall↗

arXiv2022

Complex Supersymmetry in Graphene

This work analyzes monolayer graphene in external electromagnetic fields, which is described by the Dirac equation with minimal coupling. Supersymmetric quantum mechanics allows building new Dirac equations with modified magnetic fields. Here, we will use complex factorization energies and iterate the method in order to arrive at Hermitian graphene Hamiltonians. Finally, we compare these results with the matrix supersymmetric quantum mechanics approach.

cond-mat.mes-hall↗

arXiv2019

Dirac electron in graphene with magnetic fields arising from first-order intertwining operators

The behaviour of a Dirac electron in graphene, under magnetic fields which are orthogonal to the layer, is studied. The initial problem is reduced to an equivalent one, where two one-dimensional Schrödinger Hamiltonians $H^{\pm}$ are intertwined by a first order differential operator. Special magnetic field are initially chosen, in order that $V^{\pm}$ will be shape invariant exactly solvable potentials. When looking for more general first order operators, intertwining $H^-$ with a non-necessarily shape invariant Hamiltonian, new magnetic fields associated also to analytic solutions will be generated. The iteration of this procedure is as well discussed.

quant-ph↗

arXiv2019

Polynomial Heisenberg algebras, multiphoton coherent states and geometric phases

In this paper we will realize the polynomial Heisenberg algebras through the harmonic oscillator. We are going to construct then the Barut-Girardello coherent states, which coincide with the so-called multiphoton coherent states, and we will analyze the corresponding Heisenberg uncertainty relation and Wigner distribution function for some particular cases. We will show that these states are intrinsically quantum and cyclic, with a period being a fraction of the oscillator period. The associated geometric phases will be as well evaluated.

math-ph↗

arXiv2018

Darboux transformations and second order difference equations

In this paper we implement the Darboux transformation, as well as an analogue of Crum's theorem, for a discrete version of Schrödinger equation. The technique is based on the use of first order operators intertwining two difference operators of second order. This method, which has been applied successfully for differential cases, leads also to interesting non trivial results in the discrete case. The technique allows us to construct the solutions for a wide class of difference Schrödinger equations. The exact solutions for some special potentials are also found explicitly.

math.DS↗

arXiv2017

Nonlinear supercoherent states and geometric phases for the supersymmetric harmonic oscillator

Nonlinear supercoherent states, which are eigenstates of nonlinear deformations of the Kornbluth-Zypman annihilation operator for the supersymmetric harmonic oscillator, will be studied. They turn out to be expressed in terms of nonlinear coherent states, associated to the corresponding deformations of the standard annihilation operator. We will discuss as well the Heisenberg uncertainty relation for a special particular case, in order to compare our results with those obtained for the Kornbluth-Zypman linear supercoherent states. As the supersymmetric harmonic oscillator executes an evolution loop, such that the evolution operator becomes the identity at a certain time, thus the linear and nonlinear supercoherent states turn out to be cyclic and the corresponding geometric phases will be evaluated.

math-ph↗

arXiv2016

SUSY partners of the truncated oscillator, Painlevé transcendents and Bäcklund transformations

In this work the supersymmetric technique is applied to the truncated oscillator to generate Hamiltonians ruled by second and third-order polynomial Heisenberg algebras, which are connected to the Painlevé IV and Painlevé V equations respectively. The aforementioned connection is exploited to produce particular solutions to both non-linear differential equations and the Bäcklund transformations relating them.

math-ph↗

arXiv2014

Painlevé IV Coherent States

A simple way to find solutions of the Painlevé IV equation is by identifying Hamiltonian systems with third-order differential ladder operators. Some of these systems can be obtained by applying supersymmetric quantum mechanics (SUSY QM) to the harmonic oscillator. In this work, we will construct families of coherent states for such subset of SUSY partner Hamiltonians which are connected with the Painlevé IV equation. First, these coherent states are built up as eigenstates of the annihilation operator, then as displaced versions of the extremal states, both involving the third-order ladder operators, and finally as extremal states which are also displaced but now using the so called linearized ladder operators. To each SUSY partner Hamiltonian corresponds two families of coherent states: one inside the infinite subspace associated with the isospectral part of the spectrum and another one in the finite subspace generated by the states created through the SUSY technique.

math-ph↗

arXiv2013

Supersymmetric partners of the harmonic oscillator with an infinite potential barrier

Supersymmetry transformations of first and second order are used to generate Hamiltonians with known spectra departing from the harmonic oscillator with an infinite potential barrier. It is studied also the way in which the eigenfunctions of the initial Hamiltonian are transformed. The first and certain second order supersymmetric partners of the initial Hamiltonian possess third-order differential ladder operators. Since systems with this kind of operators are linked with the Painlevé IV equation, several solutions of this non-linear second-order differential equation will be simply found.

math-ph↗

arXiv2013

Supersymmetric quantum mechanics and Painleve equations

In these lecture notes we shall study first the supersymmetric quantum mechanics (SUSY QM), specially when applied to the harmonic and radial oscillators. In addition, we will define the polynomial Heisenberg algebras (PHA), and we will study the general systems ruled by them: for zero and first order we obtain the harmonic and radial oscillators, respectively; for second and third order PHA the potential is determined by solutions to Painleve IV (PIV) and Painleve V (PV) equations. Taking advantage of this connection, later on we will find solutions to PIV and PV equations expressed in terms of confluent hypergeometric functions. Furthermore, we will classify them into several solution hierarchies, according to the specific special functions they are connected with.

math-ph↗

arXiv2012

Harmonic Oscillator SUSY Partners and Evolution Loops

Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. If applied to the harmonic oscillator, a family of Hamiltonians ruled by polynomial Heisenberg algebras is obtained. In this paper it will be shown that the SUSY partner Hamiltonians of the harmonic oscillator can produce evolution loops. The corresponding geometric phases will be as well studied.

quant-ph↗

arXiv2011

Supersymmetric Quantum Mechanics and Painlevé IV Equation

As it has been proven, the determination of general one-dimensional Schrödinger Hamiltonians having third-order differential ladder operators requires to solve the Painlevé IV equation. In this work, it will be shown that some specific subsets of the higher-order supersymmetric partners of the harmonic oscillator possess third-order differential ladder operators. This allows us to introduce a simple technique for generating solutions of the Painlevé IV equation. Finally, we classify these solutions into three relevant hierarchies.

math-ph↗

arXiv1994

Geometric Phases and Mielnik's Evolution Loops

The cyclic evolutions and associated geometric phases induced by time-independent Hamiltonians are studied for the case when the evolution operator becomes the identity (those processes are called {\it evolution loops}). We make a detailed treatment of systems having equally-spaced energy levels. Special emphasis is made on the potentials which have the same spectrum as the harmonic oscillator potential (the generalized oscillator potentials) and on their recently found coherent states.

hep-th↗