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  • 1
    Electronic Resource
    Electronic Resource
    Spectral and stability aspects of quantum chaos (1991)
    Woodbury, NY : American Institute of Physics (AIP)
    Chaos 1 (1991), S. 114-121 
    Details
    ISSN: 1089-7682
    Source: AIP Digital Archive
    Topics: Physics
    Notes: The relation between the spectrum of a generalized quasienergy operator and the stability of quantum systems driven by quasiperiodic time-dependent forces is discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.165809
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  • 2
    Electronic Resource
    Electronic Resource
    A version of Thirring's approach to the Kolmogorov–Arnold–Moser theorem for quadratic Hamiltonians with degenerate twist (1998)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 39 (1998), S. 5856-5865 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We give a proof of the Kolmogorov–Arnold–Moser (KAM) theorem on the existence of invariant tori for weakly perturbed Hamiltonian systems, based on Thirring's approach for Hamiltonians that are quadratic in the action variables. The main point of this approach is that the iteration of canonical transformations on which the proof is based stays within the space of quadratic Hamiltonians. We show that Thirring's proof for nondegenerate Hamiltonians can be adapted to Hamiltonians with degenerate twist. This case, in fact, drastically simplifies Thirring's proof. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532599
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  • 3
    Electronic Resource
    Electronic Resource
    Stability of oscillators driven by ergodic processes (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 628-645 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The evolution of a quantum harmonic oscillator, forced by a stationary ergodic process, is considered. Under suitable conditions on the "dynamics'' of this process, it is shown that generically the spectrum of the associated quasienergy operator is continuous. These results also apply to kicked harmonic oscillators where the amplitudes of the kicks are governed by a stationary ergodic process.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530657
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  • 4
    Electronic Resource
    Electronic Resource
    A classification of Fokker-Planck models and the small and large noise asymptotics (1985)
    Springer
    Journal of statistical physics 40 (1985), S. 147-165 
    Details
    ISSN: 1572-9613
    Keywords: Fokker-Planck equation ; drift velocity ; detailed balance ; nonequilibrium
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Then-variable Fokker-Planck equation can be written in an equivalent form as a system ofn + 1 first-order equations by introducing as auxiliary variables the components of the drift velocityR l. The stationary state defines a stationaryR uniquely, which allows an intrinsic classification of the stationary states in terms of the properties ofR, without reference to detailed balance. This representation is very appropriate for the study of questions such as the existence of stationary states and their small and large noise asymptotics, as well as for the construction of models having some specified behavior.R provides also a classification of the dynamics, which corresponds to the hermiticity properties of the associated eigenvalue problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01010530
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  • 5
    Electronic Resource
    Electronic Resource
    An Approximate KAM-Renormalization-Group Scheme for Hamiltonian Systems (1999)
    Springer
    Journal of statistical physics 94 (1999), S. 241-251 
    Details
    ISSN: 1572-9613
    Keywords: KAM theory ; renormalization group ; invariant tori ; nontrivial fixed point
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We construct an approximate renormalization scheme for Hamiltonian systems with two degrees of freedom. This scheme is a combination of Kolmogorov–Arnold–Moser (KAM) theory and renormalization-group techniques. It makes the connection between the approximate renormalization procedure derived by Escande and Doveil and a systematic expansion of the transformation. In particular, we show that the two main approximations, consisting in keeping only the quadratic terms in the actions and the two main resonances, keep the essential information on the threshold of the breakup of invariant tori.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1004519514702
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  • 6
    Electronic Resource
    Electronic Resource
    Master equations for subordinated processes (1990)
    Springer
    Journal of statistical physics 60 (1990), S. 413-444 
    Details
    ISSN: 1572-9613
    Keywords: Markov jump processes ; master equation ; Levy flight
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study Markov jump processes constructed by subordination of diffusion processes. The procedure can be viewed as a randomization or a coarse graining of time. We construct the master equation for the cases of finite and infinite total jump rates, and give a collection of explicitly solvable examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01314929
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  • 7
    Electronic Resource
    Electronic Resource
    Diffusive energy growth in classical and quantum driven oscillators (1991)
    Springer
    Journal of statistical physics 62 (1991), S. 793-817 
    Details
    ISSN: 1572-9613
    Keywords: Time dependent Hamiltonian ; diffusive energy growth ; quantum chaos ; harmonic oscillator
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study the long-time stability of oscillators driven by time-dependent forces originating from dynamical systems with varying degrees of randomness. The asymptotic energy growth is related to ergodic properties of the dynamical system: when the autocorrelation of the force decays sufficiently fast one typically obtains linear diffusive growth of the energy. For a system with good mixing properties we obtain a stronger result in the form of a central limit theorem. If the autocorrelation decays slowly or does not decay, the behavior can depend on subtle properties of the particular model. We study this dependence in detail for a family of quasiperiodic forces. The solution involves the analysis of a small-denominator problem that can be treated by fairly elementary methods. In the special case of a periodic force the quantum stability problem can be expressed in terms of spectral properties of the Floquet operator. In the presence of resonances the spectrum is absolutely continuous. We find explicitly the eigenvalues and eigenfunctions for the nonresonant case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01017984
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  • 8
    Electronic Resource
    Electronic Resource
    Floquet spectrum for two-level systems in quasiperiodic time-dependent fields (1992)
    Springer
    Journal of statistical physics 68 (1992), S. 271-310 
    Details
    ISSN: 1572-9613
    Keywords: Quasiperiodic ; Floquet operator ; quasienergy ; quantum chaos ; KAM ; Rabi oscillations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study the time evolution ofN-level quantum systems under quasiperiodic time-dependent perturbations. The problem is formulated in terms of the spectral properties of a quasienergy operator defined in an enlarged Hilbert space, or equivalently of a generalized Floquet operator. We discuss criteria for the appearance of pure point as well as continuous spectrum, corresponding respectively to stable quasiperiodic dynamics and to unstable chaotic behavior. We discuss two types of mechanisms that lead to instability. The first one is due to near resonances, while the second one is of topological nature and can be present for arbitrary ratios between the frequencies of the perturbation. We treat explicitly an example of this type. The stability of the pure point spectrum under small perturbations is proven using KAM techniques.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01048846
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  • 9
    Electronic Resource
    Electronic Resource
    Melnikov's criterion for nondifferentiable weak-noise potentials (1986)
    Springer
    Journal of statistical physics 42 (1986), S. 573-585 
    Details
    ISSN: 1572-9613
    Keywords: Weak-noise ; Fokker-Planck ; Hamilton-Jacobi ; Melnikov function ; nonequilibrium ; nondifferentiable potential
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The stationary probability density of Fokker-Planck models with weak noiseη is asymptotically of the form exp[−1 /ηϕ(q)]. Ifϕ is smooth, it satisfies a Hamilton-Jacobi equation at zero energy and can be interpreted as the action of an associated Hamiltonian system. Under this assumption,ϕ has the properties of a Liapounov function, and can be used, e.g., as a thermodynamic potential in nonequilibrium steady states. We consider systems having several attractors and show, by applying Melnikov's method to the associated Hamiltonian, that in generalϕ is not differentiable. A small perturbation of a model with differentiableϕ leads to a nondifferentiable ϕ. The method is illustrated on a model used in the treatment of the unstable mode in a laser.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01127729
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  • 10
    Electronic Resource
    Electronic Resource
    Algebraic time-reversal operation (1999)
    Springer
    The European physical journal 5 (1999), S. 257-265 
    Details
    ISSN: 1434-6079
    Keywords: PACS. 03.65.Fd Algebraic methods - 31.15.Hz Group theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract: We analyze the implementation of the time-reversal (TR) transformation in the algebraic approach to tetrahedral local molecules through the chain of groups . We determine the general form of the TR operation using a purely algebraic realization, based exclusively on the requirement that the irreducible representations must not be changed under the time inversion symmetry. As a result we can determine the TR behavior of purely algebraic operators.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s100500050284
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