Related papers: Partial integrability of the anharmonic oscillator
Here we consider the anharmonic oscillator that is a dynamical system given by $y_{xx}+\delta y^{n}=0$. We demonstrate that to this equation corresponds a new example of a superintegrable two-dimensional metric with a linear and a…
The paper introduces a simple quantum model to calculate in a general way allowed frequencies and energy levels of the anharmonic oscillator. The theoretical basis of the approach has been introduced in two early papers aimed to infer the…
In this paper, we consider a generalized second order nonlinear ordinary differential equation of the form $\ddot{x}+(k_1x^q+k_2)\dot{x}+k_3x^{2q+1}+k_4x^{q+1}+\lambda_1x=0$, where $k_i$'s, $i=1,2,3,4$, $\lambda_1$ and $q$ are arbitrary…
We analyze a simple example of wave equation with a time-dependent damping term, whose coefficient decays at infinity at the scale-invariant rate and includes an oscillatory component that is integrable but not absolutely integrable. We…
We consider a damped impact oscillator subject to the action of a biharmonic force. The conditions for the existence and stability of almost periodic resonance solutions are investigated.
Sufficient oscillation conditions involving $\limsup $ and $\liminf $ for first-order differential equations with several non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative…
We investigate symmetric oscillators, and in particular their quantization, by employing semiclassical and quantum phase functions introduced in the context of Liouville-Green transformations of the Schr\"{o}dinger equation. For anharmonic…
The in-in formalism and its influence functional generalization are widely used to describe the out-of-equilibrium dynamics of unitary and open quantum systems, respectively. In this paper, we build on these techniques to develop an…
The behaviour of the interaction of the induced electric dipole moment of an atom with a uniform magnetic field and a non-uniform electric field are investigated in a rotating reference frame. An interesting aspect of this interaction is…
In this article, we formulate the study of the unitary time evolution of systems consisting of an infinite number of uncoupled time-dependent harmonic oscillators in mathematically rigorous terms. We base this analysis on the theory of a…
In this paper, it is proposed a quantization procedure for the one-dimensional harmonic oscillator with time-dependent frequency, time-dependent driven force, and time-dependent dissipative term. The method is based on the construction of…
For the model of a linearly driven quantum anharmonic oscillator, the role of damping is investigated. We compare the position of the stable points in phase space obtained from a classical analysis to the result of a quantum mechanical…
In this paper we give a general solution to the problem of the damped harmonic oscillator under the influence of an arbitrary time-dependent external force. We employ simple methods accessible for beginners and useful for undergraduate…
In recent papers by the authors (S.~Motonaga and K.~Yagasaki, Obstructions to integrability of nearly integrable dynamical systems near regular level sets, submitted for publication, and K.~Yagasaki, Nonintegrability of nearly integrable…
This is the third article in a series of three papers on the resonance energy levels of anharmonic oscillators. Whereas the first two papers mainly dealt with double-well potentials and modifications thereof [see J. Zinn-Justin and U. D.…
In this work, we present the analytical approach to the evaluation of the conditional measure Wiener path integral. We consider the time-dependent model parameters. We find the differential equation for the variable, determining the…
We use the Liouville-von Neumann (LvN) approach to study the dynamics and the adiabaticity of a time-dependent driven anharmonic oscillator as an eample of non-equilibrium quantum dynamics. We show that the adiabaticity is minimally broken…
The time dependent entropy (or Leipnik's entropy) of harmonic and damped harmonic oscillators is extensively investigated by using time dependent wave function obtained by the Feynman path integral method. Our results for simple harmonic…
We consider the nonlinear Duffing oscillator in presence of fractional damping which is characteristic in different physical situations. The system is studied with a smaller and larger damping parameter value, that we call the underdamped…
The von Neumann interaction between a particle and an apparatus, both of arbitrary mass, has been considered in the measurement of the position of a simple harmonic oscillator acted on by an external force. When the measurement has finite…