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AN INTEGRAL EXPRESSION OF EXACT SOLUTION FOR HARMONIC OSCILLATOR WITH TIME-VARYING FREQUENCY

Li Li-xiang,Guo Guang-can,

中国物理 B , 1998,
Abstract: Using Feynman's method of disentangling operators, an exact expression of evolution operator of harmonic oscillator with time-varying frequency is obtained. The parameters in this expression are related to the time-varying frequency directly. Some implications are further investigated, including general form of linear and quadratic invariants and evolution of Wigner function.
Band invariants for perturbations of the harmonic oscillator  [PDF]
Victor Guillemin,Alejandro Uribe,Zuoqin Wang
Mathematics , 2011,
Abstract: We study the direct and inverse spectral problems for semiclassical operators of the form $S = S_0 +\h^2V$, where $S_0 = \frac 12 \Bigl(-\h^2\Delta_{\bbR^n} + |x|^2\Bigr)$ is the harmonic oscillator and $V:\bbR^n\to\bbR$ is a tempered smooth function. We show that the spectrum of $S$ forms eigenvalue clusters as $\h$ tends to zero, and compute the first two associated "band invariants". We derive several inverse spectral results for $V$, under various assumptions. In particular we prove that, in two dimensions, generic analytic potentials that are even with respect to each variable are spectrally determined (up to a rotation).
Exact invariants and adiabatic invariants of the singular Lagrange system

Chen Xiang-Wei,Li Yan-Min,

中国物理 B , 2003,
Abstract: Based on the theory of symmetries and conserved quantities of the singular Lagrange system, the perturbations to the symmetries and adiabatic invariants of the singular Lagrange systems are discussed. Firstly, the concept of higher-order adiabatic invariants of the singular Lagrange system is proposed. Then, the conditions for the existence of the exact invariants and adiabatic invariants are proved, and their forms are given. Finally, an example is presented to illustrate these results.
Exact invariants and adiabatic invariants of dynamical system of relative motion
Exact invariants and adiabatic invariantsof dynamical system of relative motion

Chen Xiang-Wei,Wang Xin-Min,Wang Ming-Quan,
陈向炜
,王新民,王明泉

中国物理 B , 2004,
Abstract: Based on the theory of symmetries and conserved quantities, the exact invariants and adiabatic invariants of a dynamical system of relative motion are studied. The perturbation to symmetries for the dynamical system of relative motion under small excitation is discussed. The concept of high-order adiabatic invariant is presented, and the form of exact invariants and adiabatic invariants as well as the conditions for their existence are given. Then the corresponding inverse problem is studied.
On the exact discretization of the classical harmonic oscillator equation  [PDF]
Jan L. Cieslinski
Computer Science , 2009,
Abstract: We discuss the exact discretization of the classical harmonic oscillator equation (including the inhomogeneous case and multidimensional generalizations) with a special stress on the energy integral. We present and suggest some numerical applications.
THE EXACT SOLUTION AND BERRY''S PHASE FOR THE GENERALIZED TIME-DEPENDENT HARMONIC OSCILLATOR
广义含时谐振子的精确解和Berry相因数

GAO XIAO-CHUN,XU JIN-BO,QIAN TIE-ZHENG,
高孝纯
,许晶波,钱铁铮

物理学报 , 1991,
Abstract: In this paper, we find the exact solution for the generalized time-dependent harmonic oscillator by making use of the Lewis-Riesenfeld theory. Then, the adiabatic asymptotic limit of the exact solution is discussed and the Berry's phase factor for the oscillator obtained. We proceed to use the exact solution to construct the coherent state and calculate the corresponding classical Hannay angle.
EXACT AND ADIABATIC INVARIANTS OF FIRST-ORDER LAGRANGE SYSTEMS

Chen Xiang-wei,Shang Mei,Mei Feng-xiang,

中国物理 B , 2001,
Abstract: A system of first-order differential equations is expressed in the form of first-order Lagrange equations. Based on the theory of symmetries and conserved quantities of first-order Lagrange systems, the perturbation to the symmetries and adiabatic invariants of first-order Lagrange systems are discussed. Firstly, the concept of higher-order adiabatic invariants of the first-order Lagrange system is proposed. Then, conditions for the existence of the exact and adiabatic invariants are proved, and their forms are given. Finally, an example is presented to illustrate these results.
Amplitude and phase representation of quantum invariants for the time dependent harmonic oscillator  [PDF]
M. Fernandez Guasti,H. Moya-Cessa
Physics , 2002, DOI: 10.1103/PhysRevA.67.063803
Abstract: The correspondence between classical and quantum invariants is established. The Ermakov Lewis quantum invariant of the time dependent harmonic oscillator is translated from the coordinate and momentum operators into amplitude and phase operators. In doing so, Turski's phase operator as well as Susskind-Glogower operators are generalized to the time dependent harmonic oscillator case. A quantum derivation of the Manley-Rowe relations is shown as an example.
Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential  [PDF]
Dae-Yup Song
Physics , 2000, DOI: 10.1103/PhysRevA.62.014103
Abstract: The unitary operator which transforms a harmonic oscillator system of time-dependent frequency into that of a simple harmonic oscillator of different time-scale is found, with and without an inverse-square potential. It is shown that for both cases, this operator can be used in finding complete sets of wave functions of a generalized harmonic oscillator system from the well-known sets of the simple harmonic oscillator. Exact invariants of the time-dependent systems can also be obtained from the constant Hamiltonians of unit mass and frequency by making use of this unitary transformation. The geometric phases for the wave functions of a generalized harmonic oscillator with an inverse-square potential are given.
Non-Adiabatic Solution to the Time Dependent Quantum Harmonic Oscillator  [PDF]
C. A. M. de Melo,B. M. Pimentel,J. A. Ramirez
Physics , 2010,
Abstract: Using Schwinger Variational Principle we solve the problem of quantum harmonic oscillator with time dependent frequency. Here, we do not take the usual approach which implicitly assumes an adiabatic behavior for the frequency. Instead, we propose a new solution where the frequency only needs continuity in its first derivative or to have a finite set of removable discontinuities.
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