arXiv subjects

L. M. Nieto

Publications and source records attributed to L. M. Nieto.

At least 37 records · Page 2Linked to original sources

arXiv2023

Polar coherent states in bilayer graphene under a constant uniform magnetic field

Symmetries associated with the Hamiltonian describing bilayer graphene subjected to a constant magnetic field perpendicular to the plane of the bilayer are calculated using polar coordinates. These symmetries are then applied to explain some fundamental properties, such as the spectrum and the integer pseudo-spin character of the eigenfunctions. The probability and current densities of the bilayer Hamiltonian have also been calculated in polar coordinates and shown to be gauge invariant and scalar under generalized rotations. We also define appropriate coherent states of this system as eigenfunctions, with complex eigenvalues, of a suitable chose annihilation operator. In this framework, symmetries are also useful to show the meaning of the complex eigenvalue in terms of expected values. The local current density of these coherent states is shown to exhibit a kind of radial component interference effect, something that has gone unnoticed until now. Some of these results that have just been exposed are graphically illustrated throughout the manuscript.

cond-mat.mes-hall↗

arXiv2023

Inclusion of radiation in the CCM approach of the $ϕ^4$ model

We present an effective Lagrangian for the $ϕ^4$ model that includes radiation modes as collective coordinates. The coupling between these modes to the discrete part of the spectrum, i.e., the zero mode and the shape mode, gives rise to different phenomena which can be understood in a simple way in our approach. In particular, the energy transfer between radiation, translation and shape modes is carefully investigated in the single-kink sector. Finally, we also discuss the inclusion of radiation modes in the study of oscillons. This leads to relevant phenomena such as the oscillon decay and the kink-antikink creation.

hep-th↗

arXiv2022

Wobbling kinks in a two-component scalar field theory: Interaction between shape modes

In this paper the interaction between the shape modes of the wobbling kinks arising in the family of two-component MSTB scalar field theory models is studied. The spectrum of the second order small kink fluctuation in this model has two localized vibrational modes associated to longitudinal and orthogonal fluctuations with respect to the kink orbit. It has been found that the excitation of the orthogonal shape mode immediately triggers the longitudinal one. In the first component channel the kink emits radiation with twice the orthogonal wobbling frequency (not the longitudinal one as happens in the $ϕ^4$-model). The radiation emitted in the second component has two dominant frequencies: one is three times the frequency of the orthogonal wobbling mode and the other is the sum of the frequencies of the longitudinal and orthogonal vibration modes. This feature is explained analytically using perturbation expansion theories.

hep-th↗

arXiv2022

Massive and massless two-dimensional Dirac particles in electric quantum dots

In this work we investigate the confining properties of charged particles of a Dirac material in the plane subject to an electrostatic potential well, that is, in an electric quantum dot. Our study focuses on the effect of mass and angular momenta on such confining properties. To have a global picture of confinement, both bound and resonance states are considered. The resonances will be examined by means of the Wigner time delay of the scattering states, as well as through the complex eigenvalues of outgoing states in order to show that they are physically meaningful. By tuning the potential intensity of the well, electron captures and atomic collapses are observed for critical values. In these processes, the bound states of the discrete spectrum become resonances of the continuous spectrum or vice versa. For massive charges, the atomic collapse phenomenon keeps the number of bound levels in the quantum dot below a maximum value. In the massless case, the bound states have zero energy and occur only for some discrete values of the potential depth, as is known. We also show that although the intensity of the resonances for massive particles is not significantly influenced by angular momenta, on the contrary, for massless particles they are quite sensitive to angular momenta, as it is the case of graphene.

quant-ph↗

arXiv2021

Asymmetric scattering between kinks and wobblers

The asymmetric scattering between wobblers and kinks in the standard $ϕ^4$ model is numerically investigated in two different scenarios. First, the collision between wobblers with opposite phase is analyzed. Here, a destructive interference between the shape modes of the colliding wobblers takes place at the impact time. The second scenario involves the scattering between a wobbler and an (unexcited) kink. In this case the energy transfer from the wobbler to the kink can be examined. The dependence of the final velocities and wobbling amplitudes of the scattered wobblers on the collision velocity and on the initial wobbling amplitude is discussed. Both situations lead to very different fractal structures in the velocity diagrams.

hep-th↗

arXiv2021

Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian

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 extensions. There are essentially two big groups of extensions. In one, the ground state has strictly positive energy. On the other, either the ground state has zero or negative energy. In the present paper, we show that each of the extensions belonging to the first group (energy of ground state strictly positive) has an infinite sequence of supersymmetric partners, such that the $\ell$-th order partner differs in one energy level from both the $(\ell-1)$-th and the $(\ell+1)$-th order partners. In general, the eigenvalues for each of the self-adjoint extensions of $-d^2/dx^2$ come from a transcendental equation and are all infinite. For the case under our study, we determine the eigenvalues, which are also infinite, {all the extensions have a purely discrete spectrum,} and their respective eigenfunctions for all of its $\ell$-th supersymmetric partners of each extension.

math-ph↗

arXiv2021

The momentum distribution of two bosons in one dimension with infinite contact repulsion in harmonic trap gets analytical

For a harmonically trapped system consisting of two bosons in one spatial dimension with infinite contact repulsion (hard core bosons), we derive an expression for the one-body density matrix $ρ_B$ in terms of centre of mass and relative coordinates of the particles. The deviation from $ρ_F$, the density matrix for the two fermions case, can be clearly identified. Moreover, the obtained $ρ_B$ allows us to derive a closed form expression of the corresponding momentum distribution $n_{B}(p)$. We show how the result deviates from the noninteracting fermionic case, the deviation being associated to the short range character of the interaction. Mathematically, our analytical momentum distribution is expressed in terms of one and two variables confluent hypergeometric functions. Our formula satisfies the correct normalization and possesses the expected behavior at zero momentum. It also exhibits the high momentum $1/p^4 $ tail with the appropriate Tan's coefficient. Numerical results support our findings.

physics.atom-ph↗

arXiv2021

The Schrödinger particle on the half-line with an attractive $δ$-interaction: bound states and resonances

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 $-λδ(x-x_0)$ for any fixed value of the magnitude of the coupling constant. We also investigate the $λ$-dependence of both eigenvalues for any fixed value of $x_0$. Furthermore, we show that both systems exhibit resonances as poles of the analytic continuation of the resolvent. These results will be connected with the study of the ground state energy of two remarkable three-dimensional self-adjoint operators, studied in depth in Albeverio's monograph, perturbed by an attractive $δ$-distribution supported on the spherical shell of radius $r_0$.

math-ph↗

arXiv2021

Scattering between wobbling kinks

In this paper the scattering between a wobbling kink and a wobbling antikink in the standard $ϕ^4$ model is numerically investigated. The dependence of the final velocities, wobbling amplitudes and frequencies of the scattered kinks on the collision velocity and on the initial wobbling amplitude is discussed. The fractal structure becomes more intricate due to the emergence of new resonance windows and the splitting of those arising in the non-excited kink scattering. Outside this phase the final wobbling amplitude exhibits a linear dependence of the collision velocity whereas the final frequency is a decreasing function. By contrast these magnitudes are almost independent of the initial wobbling amplitude.

hep-th↗

arXiv2020

Band spectra of periodic hybrid $δ$-$δ^\prime$ structures

We present a detailed study of a generalised one-dimensional Kronig-Penney model using $δ\text{-}δ'$ 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 $δ\text{-}δ'$ potentials standing at the linear lattice nodes. This case will be known throughout the paper as the one-species hybrid Dirac comb. We investigate the consequences of adding the $δ'$ interaction to the Dirac comb by comparing the band spectra and the density of states of pure Dirac-$δ$ combs and one-species hybrid Dirac combs. Secondly we study the quantum system that arises when the periodic potential is the one obtained from the superposition of two one-species hybrid Dirac combs displaced one with respect to the other and with different couplings. The latter will be known as the two-species hybrid Dirac comb. One of the most remarkable results is the appearance of a curvature change in the band spectrum when the $δ'$ couplings are above a critical value.

math-ph↗

arXiv2020

Tavis-Cummings models and their quasi-exactly solvable Schrödinger Hamiltonians

We study in detail the relationship between the Tavis-Cummings Hamiltonian of quantum optics and a family of quasi-exactly solvable Schrödinger equations. The connection between them is stablished through the biconfluent Heun equation. We found that each invariant $n$-dimensional subspace of Tavis-Cummings Hamiltonian corresponds either to $n$ potentials, each with one known solution, or to one potential with $n$-known solutions. Among these Schrödinger potentials appear the quarkonium and the sextic oscillator.

quant-ph↗

arXiv2020

An approximation to the Woods-Saxon potential based on a contact interaction

We study a non-relativistic particle subject to a three-dimensional spherical potential consisting of a finite well and a radial $δ$-$δ'$ 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 Hamiltonian. Since this model admits exact solutions for the wave function, we are able to characterize and calculate the number of bound states. We also extend some well-known properties of certain spherically symmetric potentials and describe the resonances, defined as unstable quantum states. Based on the Woods-Saxon potential, this configuration is implemented as a first approximation for a mean-field nuclear model. The results derived are tested with experimental and numerical data in the double magic nuclei $^{132}$Sn and $^{208}$Pb with an extra neutron.

nucl-th↗

arXiv2018

Hyperspherical ${δ\text{-}δ^\prime}$ potentials

The spherically symmetric potential $a \,δ(r-r_0)+b\,δ' (r-r_0)$ is generalised for the $d$-dimensional space as a characterisation of a unique selfadjoint extension of the free Hamiltonian. For this extension of the Dirac delta, the spectrum of negative, zero and positive energy states is studied in $d\geq 2$, providing numerical results for the expectation value of the radius as a function of the free parameters of the potential. Remarkably, only if $d=2$ the $δ$-$δ'$ potential for arbitrary $a>0$ admits a bound state with zero angular momentum.

math-ph↗

arXiv2017

On the spectrum of the Schrödinger Hamiltonian of the one-dimensional conic oscillator perturbed by a point interaction

We decorate the one-dimensional conic oscillator $\frac{1}{2} \left[-\frac{d^{2} }{dx^{2} } + \left|x \right| \right]$ with a point impurity of either $δ$-type, or local $δ'$-type or even nonlocal $δ'$-type. All the three cases are exactly solvable models, which are explicitly solved and analysed, as a first step towards higher dimensional models of physical relevance. We analyse the behaviour of the change in the energy levels when an interaction of the type $-λ\,δ(x)$ or $-λ\,δ(x-x_0)$ is switched on. In the first case, even energy levels (pertaining to antisymmetric bound states) remain invariant with $λ$ although odd energy levels (pertaining to symmetric bound states) decrease as $λ$ increases. In the second, all energy levels decrease when the form factor $λ$ increases. A similar study has been performed for the so called nonlocal $δ'$ interaction, requiring a coupling constant renormalization, which implies the replacement of the form factor $λ$ by a renormalized form factor $β$. In terms of $β$, even energy levels are unchanged. However, we show the existence of level crossings: after a fixed value of $β$ the energy of each odd level, with the natural exception of the first one, becomes lower than the constant energy of the previous even level. Finally, we consider an interaction of the type $-aδ(x)+bδ'(x)$, and analyse in detail the discrete spectrum of the resulting self-adjoint Hamiltonian.

math-ph↗

arXiv2017

Superintegrability of the Fock-Darwin system

The Fock-Darwin system is analysed from the point of view of its symmetry properties in the quantum and classical frameworks. The quantum Fock-Darwin system is known to have two sets of ladder operators, a fact which guarantees its solvability. We show that for rational values of the quotient of two relevant frequencies, this system is superintegrable, the quantum symmetries being responsible for the degeneracy of the energy levels. These symmetries are of higher order and close a polynomial algebra. In the classical case, the ladder operators are replaced by ladder functions and the symmetries by constants of motion. We also prove that the rational classical system is superintegrable and its trajectories are closed. The constants of motion are also generators of symmetry transformations in the phase space that have been integrated for some special cases. These transformations connect different trajectories with the same energy. The coherent states of the quantum superintegrable system are found and they reproduce the closed trajectories of the classical one.

math-ph↗

arXiv2017

Heisenberg-type higher order symmetries of superintegrable systems separable in cartesian coordinates

Heisenberg-type higher order symmetries are studied for both classical and quantum mechanical systems separable in cartesian coordinates. A few particular cases of this type of superintegrable systems were already considered in the literature, but here they are characterized in full generality together with their integrability properties. Some of these systems are defined only in a region of $\mathbb R^n$, and in general they do not include bounded solutions. The quantum symmetries and potentials are shown to reduce to their superintegrable classical analogs in the $\hbar \to0$ limit.

nlin.SI↗

arXiv2017

Dipole-dipole interaction in cavity-QED: perturbative regime

We compute the interaction energies of a two-atom system placed in the middle of a perfectly reflecting planar cavity, in the perturbative regime. Explicit expressions are provided for the van der Waals potentials of two polarisable atomic dipoles as well as for the electrostatic potential of two induced dipoles. For the van der Waals potentials, several scenarios are considered, namely, a pair of atoms in their ground states, a pair of atoms both excited, and a pair of dissimilar atoms with one of them excited. In addition, the corresponding phase-shift of the two-atom wavefunction is calculated in each case. The effects of the two-dimensional confinement of the electromagnetic field by the cavity are analyzed in each scenario.

quant-ph↗

arXiv2015

Two-point one-dimensional $δ$-$δ^\prime$ interactions: non-abelian addition law and decoupling limit

In this contribution to the study of one dimensional point potentials, we prove that if we take the limit $q\to 0$ on a potential of the type $v_0δ({y})+{2}v_1δ'({y})+w_0δ({y}-q)+ {2} w_1δ'({y}-q)$, we obtain a new point potential of the type ${u_0} δ({y})+{2 u_1} δ'({y})$, when $ u_0$ and $ u_1$ are related to $v_0$, $v_1$, $w_0$ and $w_1$ by a law having the structure of a group. This is the Borel subgroup of $SL_2({\mathbb R})$. We also obtain the non-abelian addition law from the scattering data. The spectra of the Hamiltonian in the exceptional cases emerging in the study are also described in full detail. It is shown that for the $v_1=\pm 1$, $w_1=\pm 1$ values of the $δ^\prime$ couplings the singular Kurasov matrices become equivalent to Dirichlet at one side of the point interaction and Robin boundary conditions at the other side.

math-ph↗