Dimensional Oscillator Research Articles

Exact solutions of the (3+1)-dimensional DKP oscillator in Doubly Special Relativity

Exact solutions of the (3+1)-dimensional DKP oscillator in Doubly Special Relativity

Spherical analysis on the $$(2n+2)$$-dimensional oscillator group

Spherical analysis on the $$(2n+2)$$-dimensional oscillator group

Exact Green’s function for 2D dirac oscillator in constant magnetic field within curved snyder space, and its thermal properties

Following the path integral approach, and in the context of curved Snyder space, we formulate the Green function for a (1+1)-dimensional Dirac oscillator system subject to a homogeneous magnetic field. Using the radial coordinates transformation the Green function and the electron propagator are calculated. Consequently, the exact bound states and their corresponding spectral energies are extracted. Our analysis has revealed that, under specific conditions when mω¯→mωc/2 and c → V F , the behavior of the Dirac oscillator system in the presence of a uniform magnetic field within the SdS algebra closely resembles the dynamics of the monolayer graphene problem in the same algebraic framework. At high temperatures, the thermodynamic properties of the electron gas in the four cases of deformation parameters were extracted. The effect of the deformation parameters on these properties are tested, and also the limit cases for small parameters were inferred.

On the Stable Eigenvalues of Perturbed Anharmonic Oscillators in Dimension Two

We study the asymptotic behavior of the spectrum of a quantum system which is a perturbation of a spherically symmetric anharmonic oscillator in dimension 2. We prove that a large part of its eigenvalues can be obtained by Bohr-Sommerfeld quantization rule applied to the normal form Hamiltonian and also admit an asymptotic expansion at infinity. The proof is based on the generalization to the present context of the normal form approach developed in [BLM20b] (see also [PS10]) for the particular case of $T^d$.

Electronically Tunable 3D Autonomous Chaotic Oscillator Employing Single CCCFA and Its Extension to 4D

This work presents the realization of a novel electronically tuneable third-order autonomous chaotic oscillator using a current-controlled current conveyor feedback amplifier (CCCFA). The proposed circuit consists of a single CCCFA, two grounded passive capacitors, one inductor, and two diodes. The chaotic oscillator possesses smooth symmetrical sin-hyperbolic nonlinearity through two antiparallelly connected diodes. The electronically tunable intrinsic resistance at the X terminal serves as the bifurcation control parameter in the circuit and is controlled through bias current. Tuning of bias current enables the formation of different periodic and chaotic attractors. The proposed chaotic oscillator exhibits rich nonlinear dynamical behavior such as periodicity, antimonotonicity, and coexistence of attractors. Multistability is investigated through circuit-level simulation. The chaotic oscillator can easily be implemented in an integrated circuit as the configuration is simple, resistorless, and uses the minimum number of components in the count. Further, a higher dimensional (4D) chaotic oscillator is proposed as an extended circuit of the 3D chaotic oscillator and illustrated as an application in chaos encryption. Simulations are done using TSMC 180nm CMOS technology in PSpice. Experimental results are presented to verify the theoretical and simulation analyses of the proposed circuits.

Kernel controlled real-time Complex Langevin simulation

This study explores the utility of a kernel in complex Langevin simulations of quantum real-time dynamics on the Schwinger-Keldysh contour. We give several examples where we use a systematic scheme to find kernels that restore correct convergence of complex Langevin. The schemes combine prior information we know about the system and the correctness of convergence of complex Langevin to construct a kernel. This allows us to simulate up to 2β on the real-time Schwinger-Keldysh contour with the 0 + 1 dimensional anharmonic oscillator using m = 1; λ = 24, which was previously unattainable using the complex Langevin equation.

Heat transport in long-ranged Fermi-Pasta-Ulam-Tsingou-type models.

We perform a detailed study of heat transport in one-dimensional long-ranged anharmonic oscillator systems, such as the long-ranged Fermi-Pasta-Ulam-Tsingou model. For these systems, the long-ranged anharmonic potential decays with distance as a power law, controlled by an exponent δ≥0. For such a nonintegrable model, one of the recent results that has captured quite some attention is the puzzling ballisticlike transport observed for δ=2, reminiscent of integrable systems. Here, we first employ the reverse nonequilibrium molecular dynamics simulations to look closely at the δ=2 transport in three long-ranged models and point out a few problematic issues with this simulation method. Next, we examine the process of energy relaxation, and find that relaxation can be appreciably slow for δ=2 in some situations. We invoke the concept of nonlinear localized modes of excitation, also known as discrete breathers, and demonstrate that the slow relaxation and the ballisticlike transport properties can be consistently explained in terms of a novel depinning of the discrete breathers that makes them highly mobile at δ=2. Finally, in the presence of quartic pinning potentials we find that the long-ranged model exhibits Fourier (diffusive) transport at δ=2, as one would expect from short-ranged interacting systems with broken momentum conservation. Such a diffusive regime is not observed for harmonic pinning.

On the classical and quantum dynamics of a class of nonpolynomial oscillators

We consider two one dimensional nonlinear oscillators, namely (i) Higgs oscillator and (ii) a k-dependent nonpolynomial rational potential, where k is the constant curvature of a Riemannian manifold. Both the systems are of position dependent mass (PDM) form, , belonging to the quadratic Linard type nonlinear oscillators. They admit different kinds of motions at the classical level. While solving the quantum versions of the systems, we consider a generalized PDM Hamiltonian in which the ordering parameters of the mass term are treated as arbitrary. We observe that the quantum version of the Higgs oscillator is exactly solvable under appropriate restrictions of the ordering parameters, while the second nonlinear system is shown to be quasi exactly solvable using the Bethe ansatz method in which the arbitrariness of ordering parameters also plays an important role to obtain quasi-polynomial solutions. We extend the study to three dimensional generalizations of these nonlinear oscillators and obtain the exact solutions for the classical and quantum versions of the three dimensional Higgs oscillator. The three dimensional generalization of the quantum counterpart of the k-dependent nonpolynomial potential is found out to be quasi exactly solvable.

POSSIBLE ENERGY SPECTRUM OF THE TWO DIMENSIONAL SPATIAL KLEIN-GORDON OSCILLATOR EQUATION

In the present article, one had found a complete energy spectrum and eigen-function of the solution of two dimensional (2D) spatial Klein-Gordon oscillator equation which shows that its relativistic degeneracy energy is related to the eigen-function.

D-dimensional oscillators in simplicial structures: Odd and even dimensions display different synchronization scenarios

From biology to social science, the functioning of a wide range of systems is the result of elementary interactions which involve more than two constituents, so that their description has unavoidably to go beyond simple pairwise-relationships. Simplicial complexes are therefore the mathematical objects providing a faithful representation of such systems. We here present a complete theory of synchronization of $D$-dimensional oscillators obeying an extended Kuramoto model, and interacting by means of 1- and 2- simplices. Not only our theory fully describes and unveils the intimate reasons and mechanisms for what was observed so far with pairwise interactions, but it also offers predictions for a series of rich and novel behaviors in simplicial structures, which include: a) a discontinuous de-synchronization transition at positive values of the coupling strength for all dimensions, b) an extra discontinuous transition at zero coupling for all odd dimensions, and c) the occurrence of partially synchronized states at $D=2$ (and all odd $D$) even for negative values of the coupling strength, a feature which is inherently prohibited with pairwise-interactions. Furthermore, our theory untangles several aspects of the emergent behavior: the system can never fully synchronize from disorder, and is characterized by an extreme multi-stability, in that the asymptotic stationary synchronized states depend always on the initial conditions. All our theoretical predictions are fully corroborated by extensive numerical simulations. Our results elucidate the dramatic and novel effects that higher-order interactions may induce in the collective dynamics of ensembles of coupled $D$-dimensional oscillators, and can therefore be of value and interest for the understanding of many phenomena observed in nature, like for instance the swarming and/or flocking processes unfolding in three or more dimensions.

Fractional order biological snap oscillator: Analysis and control

The manuscript studies the four dimensional fractional order biological snap oscillator. Its thorough dynamical analysis is performed using dynamical tools such as phase portraits, Lyapunov and bifurcation analysis, system’s solution, stagnation points analysis etc. The impact of fractional order on the chaotic system is also discussed by plotting phase diagrams, Lyapunov and bifurcation diagrams for variable order. Considering uncertainties and disturbances trajectories are stabilized about its unique stagnation point using adaptive SMC technique. The controllers are constructed to compensate undesirable chaotic/ hyper chaotic behavior of the biological system which may give rise to fatal health pathological conditions. Also SMC technique show immediate effect in controlling of chaos in the system. Numerical simulations are performed using MATLAB.

Quasi coherent state of the Dirac oscillator

We construct quasi coherent state (QCS) of the dimensional Dirac oscillator as eigenstates of matrix annihilation operators. It has been shown that the QCS exhibits non-classical features like quadrature squeezing. It has also been found that for a general complex coherence parameter the quadratures are correlated and the QCS satisfies Schrödinger–Robertson uncertainty relation.

Splitting frequency of the (2 + 1)-dimensional Duffin-Kemmer-Petiau oscillator in an external magnetic field

We revisit the (2+1)-dimensional DKP oscillator in an external magnetic field by means of 4×4 and 6×6 representations of the DKP field, thus obtaining several cases studied in the literature. We found an splitting in the frequency of the DKP oscillator according to the spin projection that arises as an interplay between the oscillator, the magnetic field and the spin, from which the energies and the eigenfunctions are expressed in a unified way. For certain critical values of the magnetic field the oscillation in the components of spin projections 1 and −1 is cancelled. We study the thermodynamics of the canonical ensemble of the vectorial sector, where a phase transition is reported when the cancellation of the oscillation occurs. Thermodynamic potentials converge rapidly to their asymptotical expressions in the high temperature limit with the partition function symmetric under the reversion of the magnetic field.

Effects of Generalized Uncertainty Principle on the $$\mathbf (1+1) $$ Dimensional DKP Oscillator with Linear Potential

Based on $$ 3\times 3$$ irreducible representation of Duffin–Kemmer–Petiau (DKP) algebras, we obtain the bound-states energy spectrum, the wave function and the probability density of DKP oscillator with linear potential under the effect of Generalized Uncertainty Principle in the momentum space representation. In addition, the numerical results of the bound-states energy spectrum are discussed. It shows that the deformation parameter $$\beta $$ and the linear potential parameter $$\lambda $$ have non-negligible effect on the $$(1 + 1)$$ dimensional DKP oscillator system.

Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. II. Quartic and sextic anharmonicity cases

In our previous paper I (del Valle–Turbiner, 2019) a formalism was developed to study the general [Formula: see text]-dimensional radial anharmonic oscillator with potential [Formula: see text]. It was based on the Perturbation Theory (PT) in powers of [Formula: see text] (weak coupling regime) and in inverse, fractional powers of [Formula: see text] (strong coupling regime) in both [Formula: see text]-space and in [Formula: see text]-space, respectively. As a result, the Approximant was introduced — a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variational calculations, it has led to variational energies of unprecedented accuracy for cubic anharmonic oscillator. In this paper, the formalism is applied to both quartic and sextic, spherically-symmetric radial anharmonic oscillators with two term potentials [Formula: see text], [Formula: see text], respectively. It is shown that a two-parametric Approximant for quartic oscillator and a five-parametric one for sextic oscillator for the first four eigenstates used to calculate the variational energy are accurate in 8–12 figures for any [Formula: see text] and [Formula: see text], while the relative deviation of the Approximant from the exact eigenfunction is less than [Formula: see text] for any [Formula: see text].

(1+1)-dimensional Dirac oscillator with deformed algebra with minimal uncertainty in position and maximal in momentum

[Formula: see text]-dimensional Dirac oscillator with minimal uncertainty in position and maximal in momentum is investigated. To obtain energy spectrum, SUSY QM technique is applied. It is shown that the Dirac oscillator has two branches of spectrum, the first one gives the standard spectrum of the Dirac oscillator when the parameter of deformation goes to zero and the second branch does not have nondeformed limit. Maximal momentum brings an upper bound for the energy and it gives rise to the conclusion that the energy spectrum contains a finite number of eigenvalues. We also calculate partition function for the spectrum of the first type. The partition function allows us to derive thermodynamic functions of the oscillator which are obtained numerically.

GUP corrections to the Dirac oscillator in the external magnetic field

We have studied the ()-dimensional Dirac oscillator (DO) in an external magnetic field in the framework of the generalized uncertainty principle (GUP). We have calculated the perturbative corrections for the first few energy levels. We show that the infinite degeneracy of lowest Landau level is partially lifted due to GUP correction and obtain a critical value of the magnetic field for which there is no GUP correction and the DO stops oscillating.

Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. I. Generalities, cubic anharmonicity case

For the general [Formula: see text]-dimensional radial anharmonic oscillator with potential [Formula: see text] the perturbation theory (PT) in powers of coupling constant [Formula: see text] (weak coupling regime) and in inverse, fractional powers of [Formula: see text] (strong coupling regime) is developed constructively in [Formula: see text]-space and in [Formula: see text]-space, respectively. The Riccati–Bloch (RB) equation and generalized Bloch (GB) equation are introduced as ones which govern dynamics in coordinate [Formula: see text]-space and in [Formula: see text]-space, respectively, exploring the logarithmic derivative of wave function [Formula: see text]. It is shown that PT in powers of [Formula: see text] developed in RB equation leads to Taylor expansion of [Formula: see text] at small [Formula: see text] while being developed in GB equation leads to a new form of semiclassical expansion at large [Formula: see text]: it coincides with loop expansion in path integral formalism. In complementary way PT for large [Formula: see text] developed in RB equation leads to an expansion of [Formula: see text] at large [Formula: see text] and developed in GB equation leads to an expansion at small [Formula: see text]. Interpolating all four expansions for [Formula: see text] leads to a compact function (called the Approximant), which should uniformly approximate the exact eigenfunction at [Formula: see text] for any coupling constant [Formula: see text] and dimension [Formula: see text]. As a concrete application, the low-lying states of the cubic anharmonic oscillator [Formula: see text] are considered. 3 free parameters of the Approximant are fixed by taking it as a trial function in variational calculus. It is shown that the relative deviation of the Approximant from the exact ground state eigenfunction is [Formula: see text] for [Formula: see text] for coupling constant [Formula: see text] and dimension [Formula: see text] In turn, the variational energies of the low-lying states are obtained with unprecedented accuracy 7–8 s.d. for [Formula: see text] and [Formula: see text]

English

We find all the left-invariant harmonic unit vector fields on the oscillator groups. Besides, we determine the associated harmonic maps from the oscillator group into its unit tangent bundle equipped with the associated Sasaki metric. Moreover, we investigate the stability and instability of harmonic unit vector fields on compact quotients of four dimensional oscillator group G1(1).

A Lagrangian fibration of the isotropic 3-dimensional harmonic oscillator with monodromy

The isotropic harmonic oscillator in dimension 3 separates in several different coordinate systems. Separating in a particular coordinate system defines a system of three Poisson commuting integrals and, correspondingly, three commuting operators, one of which is the Hamiltonian. We show that the Lagrangian fibration defined by the Hamiltonian, the z component of the angular momentum, and a quartic integral obtained from separation in prolate spheroidal coordinates has a non-degenerate focus-focus point, and hence, non-trivial Hamiltonian monodromy for sufficiently large energies. The joint spectrum defined by the corresponding commuting quantum operators has non-trivial quantum monodromy implying that one cannot globally assign quantum numbers to the joint spectrum.