Dimensional Oscillator Research Articles

A new family of N dimensional superintegrable double singular oscillators and quadratic algebra Q(3) ⨁ so(n) ⨁ so(N-n)

We introduce a new family of N dimensional quantum superintegrable models consisting of double singular oscillators of type (n, N-n). The special cases (2,2) and (4,4) have previously been identified as the duals of 3- and 5-dimensional deformed Kepler–Coulomb systems with u(1) and su(2) monopoles, respectively. The models are multiseparable and their wave functions are obtained in (n, N-n) double-hyperspherical coordinates. We obtain the integrals of motion and construct the finitely generated polynomial algebra that is the direct sum of a quadratic algebra Q(3) involving three generators, so(n), so(N-n) (i.e. Q(3) ⨁ so(n) ⨁ so(N-n)). The structure constants of the quadratic algebra itself involve the Casimir operators of the two Lie algebras so(n) and so(N-n). Moreover, we obtain the finite dimensional unitary representations (unirreps) of the quadratic algebra and present an algebraic derivation of the degenerate energy spectrum of the superintegrable model.

Chiral phase transition and quantum revivals in graphene

We explain the dynamics of charge carriers in graphene using a two dimensional Dirac oscillator in the presence of an external magnetic field. The energy dispersion relation with linear behavior corresponds to monolayer graphene in a relativistic regime, whereas parabolic behavior appears in the case of bilayer graphene in a non-relativistic regime. We show that in the bilayer graphene model, a magnetic field-dependent energy gap exists, whereas a changing external magnetic field leads to a chiral phase transition. Our model explains the phenomenon of collapse and revivals and chiral phase transition in the presence of a magnetic field in monolayer and bilayer graphene. The displayed collapses and revivals occur due to Zitterbewegung and classical cyclotron motion.

Quantum phase transitions of the Dirac oscillator in the anti-Snyder model

We obtain exact solutions of the ($2+1$) dimensional Dirac oscillator in a homogeneous magnetic field within the anti-Snyder modified uncertainty relation characterized by a momentum cutoff ($p\ensuremath{\le}{p}_{\mathrm{max}}=1/\sqrt{\ensuremath{\beta}}$). In ordinary quantum mechanics ($\ensuremath{\beta}\ensuremath{\rightarrow}0$) this system is known to have a single left-right chiral quantum phase transition (QPT). We show that a finite momentum cutoff modifies the spectrum introducing additional quantum phase transitions. It is also shown that the presence of momentum cutoff modifies the degeneracy of the states.

NEMS With Broken T Symmetry: Graphene Based Unidirectional Acoustic Transmission Lines.

In this work we discuss the idea of one-way acoustic signal isolation in low dimensional nanoelectromechanical oscillators. We report a theoretical study showing that one-way conversion between in-phase and anti-phase vibrational modes of a double layer graphene nanoribbon is achieved by introducing spatio-temporal modulation of system properties. The required modulation length in order to reach full conversion between the two modes is subsequently calculated. Generalization of the method beyond graphene nanoribbons and realization of a NEMS signal isolator are also discussed.

Quantum phase transitions of the Dirac oscillator in a minimal length scenario

We obtain exact solutions of the (2+1) dimensional Dirac oscillator in a homogeneous magnetic field within a minimal length ($\Delta x_0=\hbar \sqrt{\beta}$), or generalised uncertainty principle (GUP) scenario. This system in ordinary quantum mechanics has a single left-right chiral quantum phase transition (QPT). We show that a non zero minimal length turns on a infinite number of quantum phase transitions which accumulate towards the known QPT when $\beta \to 0$. It is also shown that the presence of the minimal length modifies the degeneracy of the states and that in this case there exist a new class of states which do not survive in the ordinary quantum mechanics limit $\beta \to 0$.

The exact solutions of a (2 + 1)-dimensional Dirac oscillator under a magnetic field in the presence of a minimal length

Minimal length of a two-dimensional Dirac oscillator is investigated in the presence of a uniform magnetic field and illustrates the wave functions in the momentum space. The energy eigenvalues are found and the corresponding wave functions are calculated in terms of hypergeometric functions.

Dirac oscillator and nonrelativistic Snyder-de Sitter algebra

Three dimensional Dirac oscillator was considered in space with deformed commutation relations known as Snyder-de Sitter algebra. Snyder-de Sitter commutation relations give rise to appearance of minimal uncertainties in position as well as in momentum. To derive energy spectrum and wavefunctions of the Dirac oscillator, supersymmetric quantum mechanics and shape invariance technique were applied.

The Tremblay-Turbiner-Winternitz system as extended Hamiltonian

We generalize the idea of “extension of Hamiltonian systems”—developed in a series of previous articles—which allows the explicit construction of Hamiltonian systems with additional non-trivial polynomial first integrals of arbitrarily high degree, as well as the determination of new superintegrable systems from old ones. The present generalization, that we call “modified extension of Hamiltonian systems,” produces the third independent first integral for the (complete) Tremblay-Turbiner-Winternitz system, as well as for the caged anisotropic oscillator in dimension two.

The multivariate Hahn polynomials and the singular oscillator

Karlin and McGregorʼs d-variable Hahn polynomials are shown to arise in the -dimensional singular oscillator model as the overlap coefficients between bases associated with the separation of variables in Cartesian and hyperspherical coordinates. These polynomials in d discrete variables depend on real parameters and are orthogonal with respect to the multidimensional hypergeometric distribution. The focus is put on the d = 2 case for which the connection with the three-dimensional singular oscillator is used to derive the main properties of the polynomials: forward/backward shift operators, orthogonality relation, generating function, recurrence relations, bispectrality (difference equations) and explicit expression in terms of the univariate Hahn polynomials. The extension of these results to an arbitrary number of variables is presented at the end of the paper.

Quantum phase transitions in the noncommutative Dirac oscillator

We study the (2 + 1)-dimensional Dirac oscillator in a homogeneous magnetic field in the noncommutative plane. It is shown that the effect of noncommutativity is twofold: (i) momentum noncommuting coordinates simply shift the critical value $({B}_{\text{cr}})$ of the magnetic field at which the well known left-right chiral quantum phase transition takes place (in the commuting phase); (ii) noncommutativity in the space coordinates induces a new critical value of the magnetic field, ${B}_{\text{cr}}^{*}$, where there is a second quantum phase transition (right-left): this critical point disappears in the commutative limit. The change in chirality associated with the magnitude of the magnetic field is examined in detail for both critical points. The phase transitions are described in terms of the magnetization of the system. Possible applications to the physics of silicene and graphene are briefly discussed.

The chiral operators and the statistical properties of the (2+1)-dimensional Dirac oscillator in noncommutative space

We study the (2+1)-dimensional Dirac oscillator in the presence of an external uniform magnetic field in noncommutative space. Then, we obtain the energy spectrum. In addition, we consider the statistical properties of the system.

An Alternative Map from a 2 + 1 Dimensional Charged Dirac Oscillator in the Background of a Uniform Perpendicular Magnetic Field to a Quantum Optics Model

We propose an alternative map from the the 2-dimensional charged Dirac oscillator in the background of a uniform perpendicular magnetic field onto a quantum optics model which contains both Jaynes-Cummings (JC) and Anti-Jaynes-Cummings (AJC) interactions. Different from previous work, we only introduce one kind of phonons and realize a symmetrical competition which is controlled by the magnetic field. Furthermore, we find that this model behaves as a quantum phase transition when a dimensionless parameter crosses its critical value. Several characteristics of quantum phase transition are exhibited explicitly.

Dirac oscillator in perpendicular magnetic and transverse electric fields

We study (2+1) dimensional massless Dirac oscillator in the presence of perpendicular magnetic and transverse electric fields. Exact solutions are obtained and it is shown that there exists a critical magnetic field Bc such that the spectrum is different in the two regions B>Bc and B<Bc. The situation is also analyzed for the case B=Bc.

An Exactly Solvable Three Dimensional Nonlinear Quantum Oscillator and sl(2) Algebra

In this paper, first we introduce the three dimensional non-linear oscillator with a position dependent mass. In that case we start by the stationary Schrodinger equation which is generated by the three dimensional Hamiltonian. The wave function depend on three spatial variables and the usual process of variable in spherical coordinates and the wave functions will be of radial and angular solutions. We can easily solve the angular part of equation but redial part of equation will be complicated. In that case, we take advantage from sl(2) algebra and write the corresponding equation in terms of P+(r), P-(r) and P0(r) which are generators of generalized sl(2) algebra. The information of this algebra help us to obtain the energy spectrum and wave function from radial part of equation.

Using basis sets of scar functions

We present a method to efficiently compute the eigenfunctions of classically chaotic systems. The key point is the definition of a modified Gram-Schmidt procedure which selects the most suitable elements from a basis set of scar functions localized along the shortest periodic orbits of the system. In this way, one benefits from the semiclassical dynamical properties of such functions. The performance of the method is assessed by presenting an application to a quartic two-dimensional oscillator whose classical dynamics are highly chaotic. We have been able to compute the eigenfunctions of the system using a small basis set. An estimate of the basis size is obtained from the mean participation ratio. A thorough analysis of the results using different indicators, such as eigenstate reconstruction in the local representation, scar intensities, participation ratios, and error bounds, is also presented.

De Moivre's formula: are Sands' H-functions the same as Chao's?

It has often been said that there is no clear extension of the Courant-Snyder theory to coupled dynamics. In particular, one never sees an extension of the symplectic de Moivre type formula beyond one degree of freedom. In the process of analysing the existence and meaning of such a formula, I discovered quite accidentally that Sands' formalism [1], if expressed in terms of Ripken-like lattice functions germane to the full three dimensional oscillator, gives the exact same result as the more accurate Chao theory [2]. This semi-serious paper displays this elegant connection. It is based in the lattice functions of Ripken [3] as extended by Forest [4]. We review Forest's derivation from the point of view de Moivre's formula extended to three degrees of freedom.

Noncommutative Dirac oscillator in an external magnetic field

We show that (2+1)-dimensional noncommutative Dirac oscillator in an external magnetic field is mapped onto the same but with reduced angular frequency in absence of magnetic field. We construct the relativistic Landau levels by solving corresponding Dirac equation in (2+1)-dimensional noncommutative phase space. All the Landau levels become independent of noncommutative parameter for a critical value of the magnetic field. Several other interesting features along with the relevance of such models in the study of atomic transitions in a radiation field have been discussed.

Stability of linear stochastic systems via Lyapunov exponents and applications to power systems

This paper studies linear systems under sustained random perturbations. The stochastic perturbation model is given by a bounded Markov diffusion process, as it appears, e.g., in the description of load or generation uncertainties of power systems. For such systems, the Lyapunov exponents describe necessary and sufficient conditions for almost sure asymptotic stability. The paper presents several numerical methods for the computation of Lyapunov exponents and applies this methodology to linear oscillators in dimension 2 and 3, and to a one machine – infinite bus electric power system.

LYAPUNOV SPECTRA FOR KAPITZA OSCILLATOR

Here we purpose a simple but realistic model of one dimensional nonlinear Kapitza oscillator driven by sin- or cos- rapidly external oscillating periodical force. The model has a parameter 2gl=a22 of dimension one, depending on the amplitude a and frequency of modulation . Changing its value we construct phase portraits of the system in the neighbourhood of fixed points and demonstrate the changing in Lyapunov spectrum. Our purpose is to observe the behavior of system at fixed points due to the different structures of the Lyapunov spectra.DOI : http://dx.doi.org/10.22342/jims.17.1.12.39-48

2+1 Dimensional Noncommutative Dirac Oscillator and (Anti)-Jaynes-Cummings Models

We study Dirac oscillator in 2+1 dimensional noncommutative space. The model is solved exactly and the relationship with Jaynes-Cummings (JC) or anti-Jaynes-Cummings (AJC) models are investigated. We find that for a positive noncommutative parameter, there is an exact map from the 2+1 dimensional noncommutative Dirac oscillator to AJC model. However, for a negative noncommutative parameter, the noncommutative planar Dirac oscillator contains both AJC and JC terms simultaneously. Our investigation may afford a new way to study relativistic quantum mechanics models in noncommutative space by means of quantum optics method, and vice verse.