We consider a physical system which is coupled indirectly to a Markovian resevoir through an oscillator mode. This is the case, for example, in the usual model of an atomic sample in a leaky optical cavity which is ubiquitous in quantum optics. In the strong coupling limit the oscillator can be eliminated entirely from the model, leaving an effective direct coupling between the system and the resevoir. Here we provide a mathematically rigorous treatment of this limit as a weak limit of the time evolution and observables on a suitably chosen exponential domain in Fock space. The resulting effective model may contain emission and absorption as well as scattering interactions.
In this note we consider a one-dimensional quantum mechanical particle constrained by a parabolic well perturbed by a Gaussian potential. As the related Birman-Schwinger operator is trace class, the Fredholm determinant can be exploited in order to compute the modified eigenenergies which differ from those of the harmonic oscillator due to the presence of the Gaussian perturbation. By taking advantage of Wang's results on scalar products of four eigenfunctions of the harmonic oscillator, it is possible to evaluate quite accurately the two lowest-lying eigenvalues as functions of the coupling c…
The description of number of dual (quasy)-exactly solvable models with its hidden symmetry algebra has been given at different levels of analysis within the framework of generalized Kustaanheimo-Stiefel (KS)-transformations. It's shown that N-dimensionall singular oscillator and (n+1)-dimensional generalized MICZ-Kepler system are dual to each other and the duality transformation is the generalized version of the KS transformation. The solvability of the Schrodinger equation of these problems by the variables separation method is given in a double, spherical and parabolic coordinates. The quad…
In this paper, we construct corrections to the raising and lowering (i.e. ladder) operators for a quantum harmonic oscillator subjected to a polynomial type perturbation of any degree and to any order in perturbation theory. We apply our formalism to a couple of examples, namely q and p 4 perturbations, and obtain the explicit form of those operators. We also compute the expectation values of position and momentum for the above perturbations. This construction is essential for defining coherent and squeezed states for the perturbed oscillator. Furthermore, this is the first time that correctio…
The classical boundaries of the quantum singular oscillator (SO) is addressed under Weyl-Wigner phase-space and Bohmian mechanics frameworks as to comparatively evaluate phase-space and configuration space quantum trajectories as well as to compute distorting quantum fluctuations. For an engendered pure state \textit{quasi}-gaussian Wigner function that recovers the classical time evolution (at phase and configuration spaces), Bohmian trajectories are analytically obtained as to show how the SO energy and anharmonicity parameters drive the quantum regime through the so-called quantum force, wh…
Mixed mode oscillations in the Bonhoeffer-van der Pol oscillator with weak periodic perturbation
在K. Shimizu et al.（2011）我们考虑了具有非自治周期扰动的Bonhoven-van der Pol振荡器。我们表明，该文件中报道的混合模式振荡的存在下，可以使用奇异摄动的几何理论来解释。所考虑的模型可以重写为一个4维（局部3维）自治系统，在一定条件下有一个折叠的鞍结奇点，另外可以被视为一个三时间尺度之一。
Resonance in coupled nonlinear oscillators with decaying perturbations
Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences（俄罗斯科学院乌法联邦研究中心数学研究所）；Bashkir State Pedagogical University n. a. M. Akmulla（阿克穆拉巴什基尔国立师范大学）
We build a new estimate relative with Hermite functions based upon oscillatory integrals and Langer's turning point theory. From it we show that the equation $$ i \partial_t u =-\partial_x^2 u+x^2 u+ε\langle x\rangle^μ W(νx,ωt)u,\quad u=u(t,x),~x\in\mathbb R,~ 0\leq μ<\frac13,$$ can be reduced in $\mathcal H^1(\mathbb R)$ to an autonomous system for most values of the frequency vector $ω$ and $ν$, where $W(φ, θ)$ is a smooth map from $ \mathbb T^d\times \mathbb T^n$ to $\mathbb R$ and odd in $φ$.
We perturb with an additive Gaussian white noise the Hamiltonian system associated to a cubic anharmonic oscillator. The stochastic system is assumed to start from initial conditions that guarantee the existence of a periodic solution for the unperturbed equation. We write a formal expansion in powers of the diffusion parameter for the candidate solution and analyze the probabilistic properties of the sequence of the coefficients. It turns out that such coefficients are the unique strong solutions of stochastic perturbations of the famous Lamé's equation. We obtain explicit solutions in terms …
An oscillator is called isochronous if all motions have a common period. When the system is forced by a time-dependent perturbation with the same period the phenomenon of resonance may appear. We give a sufficient condition on the perturbation in order that resonance occurs when the period annulus of the isochronous oscillator is bounded. In this context, resonance means that all solutions escape from the period annulus.
We construct semiglobal singular spacetimes for the Einstein equations coupled to a massless scalar field. Consistent with the heuristic analysis of Belinskii, Khalatnikov, Lifshitz or BKL for this system, there are no oscillations due to the scalar field. (This is much simpler than the oscillatory BKL heuristics for the Einstein vacuum equations.) Prior results are due to Andersson and Rendall in the real analytic case, and Rodnianski and Speck in the smooth near-spatially-flat-FLRW case. Similar to Andersson and Rendall we give asymptotic data at the singularity, which we refer to as final d…
Let $H = H_0 + P$ denote the harmonic oscillator on $\mathbb{R}^d$ perturbed by an isotropic pseudodifferential operator $P$ of order $1$ and let $U(t) = \operatorname{exp}(- it H)$. We prove a Gutzwiller-Duistermaat-Guillemin type trace formula for $\operatorname{Tr} U(t)$. The singularities occur at times $t \in 2 π\mathbb{Z}$ and the coefficients involve the dynamics of the Hamilton flow of the symbol $σ(P)$ on the space $\mathbb{CP}^{d-1}$ of harmonic oscillator orbits of energy $1$. This is a novel kind of sub-principal symbol effect on the trace. We generalize the averaging technique of …
