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  • 1
    Electronic Resource
    Electronic Resource
    The KAM theory of systems with short range interactions, II (1984)
    Springer
    Communications in mathematical physics 96 (1984), S. 331-344 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The proof of the results on the KAM theory of systems with short range interactions, stated in [6] is completed. Estimates on the decay of the interactions generated by the iterative procedure in the KAM theorem are proved, as well as the modification of the theorems of [2–3] needed for results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01214578
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  • 2
    Electronic Resource
    Electronic Resource
    The KAM theory of systems with short range interactions, I (1984)
    Springer
    Communications in mathematical physics 96 (1984), S. 311-329 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The existence of quasiperiodic trajectories for Hamiltonian systems consisting of long chains of nearly identical subsystems, with interactions which decay rapidly with increasing distance between the interacting components, is studied. Such models are of interest in statistical mechanics. It is shown that nonergodic motions persist for much larger perturbations than prior work indicated. If the number of degrees of freedom of the system isN, the allowed perturbation decreases only as an inverse power ofN, as the number of degrees of freedom increases, rather than the inverse power ofN! which previous estimates yielded.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01214577
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  • 3
    Electronic Resource
    Electronic Resource
    Surface models with nonlocal potentials: Upper bounds (1983)
    Springer
    Communications in mathematical physics 90 (1983), S. 293-315 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The behavior of fluctuations in a class of surface models with exponentially decaying nonlocal potentials in studied. Combining a Mayer expansion with a duality transformation, we demonstrate the equivalence of these models to a class of two dimensional spin systems with nonlocal interactions. The expansions give sufficient control over the potentials to allow the fluctuations to be bounded from above by the means of complex translations in the spin representation of the model.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01205509
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    On the stability of dense point spectrum for self-adjoint operators (1986)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 71-75 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Let A be a (random) self-adjoint operator with fixed orthonormal eigenvectors, but with independently distributed random eigenvalues. [Typically, for the eigenvalue distributions, A is considered to have a dense point spectrum almost surely (a.s.).] A class of perturbations {B} is exhibited such that A+B has only point spectrum a.s. Examples are also constructed, including a rank-one perturbation B, such that A+μB has no eigenvalues (for μ≠0) a.s., despite A having dense point spectrum a.s.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527355
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  • 5
    Electronic Resource
    Electronic Resource
    Buchbesprechungen (1991)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 42 (1991), S. 966-968 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00944573
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  • 6
    Electronic Resource
    Electronic Resource
    On the elimination of non-resonance harmonics (1986)
    Springer
    Communications in mathematical physics 103 (1986), S. 351-386 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Given a weakly coupled Hamiltonian system with short range, one dimensional interactions, andany initial conditions a canonical change of variables is constructed which yields a new Hamiltonian consisting of three parts—an integrable term, a resonant term whose effects are localized in those regions of the system which give small denominators in the Kolmogorov-Arnol'd-Moser iteration scheme and a non-resonant interaction term which is very small. (In particular, much, much smaller than our original interactions.) The conditions which allow such a transformation to be constructed are independent of the number of degrees of freedom in the system, as are the estimates on the size of the various terms. Thus, if the resonances are “sparsely” distributed through the system most of the sites in the transformed Hamiltonian behave essentially like an integrable system, at least for as long a time as the trajectory of the system lies within the region where the canonical transformation is defined. In subsequent work it is shown that this time is long, and once again independent of the number of degrees of freedom in the system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01211753
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  • 7
    Electronic Resource
    Electronic Resource
    Bounds on the trajectories of a system of weakly coupled rotators (1986)
    Springer
    Communications in mathematical physics 104 (1986), S. 21-36 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract In this paper we study a classical mechanical system of weakly coupled rotators on a one-dimensional lattice. Such systems are of interest in statistical mechanics. We prove that for any site in the system there is a “large” set of initial conditions for which there exists a canonical change of variables such that the trajectory of that site, in the transformed system, is essentially indistinguishable from that of an integrable system for a long (but finite) time. Alternatively, the trajectory of this site lies very close to a torus in the phase space of the system for times very long in comparison with the typical period of the unperturbed rotators. All the estimates in this theory areindependent of the number of degrees of freedom in the system. We propose this mechanism as an explanation of certain numerical experiments.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01210790
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  • 8
    Electronic Resource
    Electronic Resource
    Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation (1997)
    Springer
    Communications in mathematical physics 190 (1997), S. 173-211 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: In this paper we describe invariant geometrical structures in the phase space of the Swift-Hohenberg equation in a neighborhood of its periodic stationary states. We show that in spite of the fact that these states are only marginally stable (i.e., the linearized problem about these states has continuous spectrum extending all the way up to zero), there exist finite dimensional invariant manifolds in the phase space of this equation which determine the long-time behavior of solutions near these stationary solutions. In particular, using this point of view, we obtain a new demonstration of Schneider's recent proof that these states are nonlinearly stable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002200050238
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  • 9
    Electronic Resource
    Electronic Resource
    Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory (1990)
    Springer
    Communications in mathematical physics 127 (1990), S. 479-528 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract In this paper the nonlinear wave equation $$u_u - u_{xx} + v(x)u(x,t) + \varepsilon u^3 (x,t) = 0$$ is studied. It is shown that for a large class of potentials,v(x), one can use KAM methods to construct periodic and quasi-periodic solutions (in time) for this equation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02104499
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  • 10
    Electronic Resource
    Electronic Resource
    The non-linear stability of front solutions for parabolic partial differential equations (1994)
    Springer
    Communications in mathematical physics 161 (1994), S. 323-334 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract For the Ginzburg-Landau equation and similar reaction-diffusion equations on the line, we show convergence ofcomplex perturbations of front solutions towards the front solutions, by exhibiting a coercive functional.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02099781
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