Coherent states of a time-dependent forced harmonic oscillator and their Aharonov-Anandan phase

A new method to construct coherent states of a time-dependent forced harmonic oscillator was given. The close relation to the classical forced oscillator and the minimum uncertainty relation were investigated. The applied periodic force (off-resonance case), in general, will attenuate the AA phase.     

On the Construction of Finite Oscillator Dictionary

A finite oscillator dictionary which has important applications in sequences designs and the compressive sensing was introduced by Gurevich, Hadani, and Sochen in [3 Gurevich , S. , Hadani , R. , Sochen , N. (2008). The finite harmonic oscillator and its applications to sequences, communication and radar. IEEE Trans. Inform. Theory 54(9):4239–4253.[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, closed formulae of the finite split oscillator dictionary 𝔖 ^s are revisited by a simple proof, the structure of nonsplit tori of the group SL(2, 𝔽 _p ) is studied, and an explicit algorithm for computing the finite nonsplit oscillator dictionary 𝔖 ^ns is described.     

Spectral asymptotics for nonsmooth perturbations of the harmonic oscillator

We obtain the spectral asymptotics of a nonsmooth perturbation of the one-dimensional harmonic oscillator. We use the technique of perturbation theory which is based on an asymptotic presentation of the part of the kernel of the nonperturbed operator resolvent in some neighborhood of an eigenvalue.     

Energy and coherence loss rates in a one-dimensional vibrational system interacting with a bath

We consider the problem of the dynamics of a Gaussian wave packet in a one-dimensional harmonic ocsillator interacting with a bath. This problem arises in many chemical and biochemical applications related to the dynamics of chemical reactions. We take the bath-oscillator interaction into account in the framework of the Redfield theory. We obtain closed expressions for Redfield-tensor elements, which allows finding the explicit time dependence of the average vibrational energy. We show that the energy loss rate is temperature-independent, is the same for all wave packets, and depends only on the spectral function of the bath. We determine the degree of coherence of the vibrational motion as the trace of the density-matrix projection on a coherently moving wave packet. We find an explicit expression for the initial coherence loss rate, which depends on the wave packet width and is directly proportional to the intensity of the interaction with the bath. The minimum coherence loss rate is observed for a “coherent” Gaussian wave packet whose width corresponds to the oscillator frequency. We calculate the limiting value of the degree of coherence for large times and show that it is independent of the structural characteristics of the bath and depends only on the parameters of the wave packet and on the temperature. It is possible that residual coherence can be preserved at low temperatures. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 130–144, October, 2007.     

Student problems with a modelling exercise

In this paper an attempt is made to outline some of the problems facing the student who undertakes a course in mathematical modelling. A case study, considered by several cohorts of students, is used as a medium for introducing the various types of obstacles the students encounter. This approach also results in a demonstration of some of the errors that can be made in operating a modelling course and the paper concludes by offering suggestions on how to overcome certain difficulties that are exposed.     

An easy trick to a periodic solution of relativistic harmonic oscillator

In this paper, the relativistic harmonic oscillator equation which is a nonlinear ordinary differential equation is investigated by Homotopy perturbation method. Selection of a linear operator, which is a part of the main operator, is one of the main steps in HPM. If the aim is to obtain a periodic solution, this choice does not work here. To overcome this lack, a linear operator is imposed, and Fourier series of sines will be used in solving the linear equations arise in the HPM. Comparison of the results, with those of resulted by Differential Transformation and Harmonic Balance Method, shows an excellent agreement.     

Regularity theory for the fractional harmonic oscillator

In this paper we develop the theory of Schauder estimates for the fractional harmonic oscillator Hσ=σ(−Δ+²|x|), 0<σ<1. More precisely, a new class of smooth functions is defined, in which we study the action of Hσ. In fact these spaces are those adapted to the operator H, hence the suited ones for this type of regularity estimates. In order to prove our results, an analysis of the interaction of the Hermite-Riesz transforms with the Hölder spaces is needed, that we believe of independent interest.     

Generalized oscillator and its coherent states

We construct a system (a generalized oscillator) that is similar to the oscillator and is related to a system of orthogonal polynomials on the real axis. We define coherent states in the Fock space associated with the generalized oscillator. In the example of the generalized oscillator related to the Gegenbauer polynomials, we prove the (super)completeness of these coherent states, i.e., we construct a measure determining a partition of unity. We present a formula that allows calculating the Mandel parameter for the constructed coherent states. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 363–380, December, 2007.     

On the sensitivity of strongly nonlinear autonomous oscillators and oscillatory waves to small perturbations

Original asymptotic solutions are determined for two autonomousdifferential equations. The application of initial conditionsfor the energy, wave number and phase shift proves to be lesscomplicated than in previous work. For the damped simple pendulum,explicit solutions demonstrate the dependence on the initialconditions. For strongly nonlinear wave packets of the Klein–Gordonequation, asymptotic solutions are compared. In both cases,the phase shift is shown to be highly sensitive to small perturbationsin the initial conditions.     

Applying reproducing kernels to the evaluation and approximation of the simple and time-dependent imaginary time harmonic oscillator path integrals

Reproduction of kernel Hilbert spaces offers an attractive setting for imaginary time path integrals, since they allow to naturally define a probability on the space of paths, which is equal to the probability associated with the paths in Feynman's path integral formulation. This study shows that if the propagator is Gaussian, its variance equals the squared norm of a linear functional on the space of paths. This can be used to rederive the harmonic oscillator propagator, as well as to offer a finite-dimensional perturbative approximation scheme for the time-dependent oscillator wave function and its ground state energy, and its bound error. The error is related to the rate of decay of the Fourier coefficients of the time-dependent part of the potential. When the rate of decay increases beyond a certain threshold, the error in the approximation over a subspace of dimension n is of order (1/n ³).     

Coulomb damping

Viscous damping is commonly discussed in beginning differential equations and physics texts but dry friction or Coulomb friction is not despite dry friction being encountered in many physical applications. One reason for avoiding this topic is that the equations involve a jump discontinuity in the damping term. In this article, we adopt an energy approach which permits a general discussion on how to investigate trajectories for second-order differential equations representing mechanical vibration models having dry friction. This approach is suitable for classroom discussion and computer laboratory investigation in beginning courses, hence introduction of dry friction need not be delayed for more advanced courses in mechanics or modelling. Our method is applied to a harmonic oscillator example and a pendulum model. One advantage of this method is that the values of the maximum deflections of a solution can be calculated without solving the differential equation either analytically or numerically, a technique that depends on only the initial conditions.     

On a Solution of the Gross--Pitaevskii Equation for a Condensate Wave Function

In this paper, we consider an approximate solution of the Gross--Pitaevskii equation in the one-dimensional case in the form of the eigenfunction of the ground state of the harmonic oscillator. A numerical experiment shows that the approximate solution obtained satisfies, with high accuracy, the Gross--Pitaevskii equation for the interatomic-interaction potential in the form of the superposition of Gaussian functions.     

Semiclassical quantization of Bohr orbits in the helium atom

We use the complex WKB-Maslov method to construct the semiclassical spectral series corresponding to the resonance Bohr orbits in the helium atom. The semiclassical energy levels represented as the Rydberg tetra series correspond to the doubly symmetrically excited states of helium-like atoms. This level series contains the Rydberg triple series reported by Richter and Wintgen in 1991, which corresponds to the Z²⁺e⁻e⁻ configuration of electrons observed by Eichmann and his collaborators in experiments on the laser excitation of the barium atom in 1992. The lower-level extrapolation of the formula obtained for the semiclassical spectrum gives the value of the ground state energy, which differs by 6% from the experimental value obtained by Bergeson and his collaborators in 1998. We also calculate the fine structure of the semiclassical spectrum due to the spin-orbit and spin-spin interactions of electrons. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 261–286, May, 2007.     

Integrable model of a two-dimensional singular spherical oscillator in a constant magnetic field

We propose an analogue of a two-dimensional singular oscillator (Smorodinsky-Winternitz oscillator) on a sphere, which is exactly solvable in the classical sense both without and with a constant magnetic field. We find the explicit classical solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 1, pp. 131–137, July, 2008.     

Discontinuous dynamics of a non-linear, self-excited, friction-induced, periodically forced oscillator

The discontinuous dynamics of a non-linear, friction-induced, periodically forced oscillator is studied. The analytical conditions for motion switchability at the velocity boundary in such a nonlinear oscillator are developed to understand the motion switching mechanism. Using such analytical conditions of motion switching, numerical predictions of the switching scenarios varying with excitation frequency and amplitude are carried out, and the parameter maps for specific periodic motions are presented. Chaotic and periodic motions are illustrated through phase planes and switching sections for a better understanding of motion mechanism of the nonlinear friction oscillator. The periodic motions with switching in such a nonlinear, frictional oscillator cannot be obtained from the traditional analysis (e.g., perturbation and harmonic balance method).     

谐和与随机噪声联合作用下Duffing振子的双峰稳态密度

研究Duffing振子在谐和与随机噪声联合作用下系统响应的双峰稳态概率密度问题．用多尺度法分离了系统的快变项,得到了系统慢变项满足的随机微分方程．用线性化方法求出了双峰稳态概率密度的表达式．数值模拟表明提出的方法是有效的．     

Analysis of bifurcations for non-linear stochastic non-smooth vibro-impact system via top Lyapunov exponent

Bifurcations are discussed by the criterion of top Lyapunov exponent. Based on the local map and Kaminski’s algorithms, a general formulation of the top Lyapunov exponents is proposed for non-linear vibro-impact oscillators with Gaussian white noise perturbation. The analytical results are verified by phase portraits and bifurcation diagrams for a classical stochastic Duffing vibro-impact oscillator. Both results are consistent.