CONTENTS § 1 Introduction § 2 Unsolved problems Conjectures Generalizations § 3 The main ideas of the proof of the exponential estimate § 4 Steepness conditions Precise statement of the main theorem § 5 Forbidden motions § 6 Resonances Resonance zones and blocks § 7 Dependence of the diameters of the discs of fast drift on the steepness of the unperturbed Hamiltonian § 8 Condition for the non-overlapping of resonances § 9 Traps in frequency systems Completion of the proof of the main theorem § 10 Statement of the lemma on the elimination of non-resonance harmonics, and of the technical lemmas used in the proof of the main theorem § 11 Remarks on the proof of the main theorem § 12 Application of the main theorem to the many-body problem References

http://iopscience.iop.org/article/10.1070/RM1977v032n06ABEH003859/pdf

An exponential estimate of the time of stability of nearly-integrable hamiltonian systems

We prove dispersive estimates for Schrodinger equations in three dimensions without making any assumptions on zero energy.

https://www.intlpress.com/site/pub/files/_fulltext/journals/dpde/2004/0001/0004/DPDE-2004-0001-0004-a001.pdf

Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I

Horospheres and the stable part of the geodesic flow

We construct a suspension of Smale's horseshoe diffeomorphism of the two-dimensional disc as a flow in an orientable three manifold. Such a suspension is natural in the sense that it occurs frequently in periodically forced nonlinear oscillators such as the Duffing equation. From this suspension we construct a knot-holder or template—a branched two-manifold with a semiflow—in such a way that the periodic orbits are isotopic to those in the full three-dimensional flow. We discuss some of the families of knotted periodic orbits carried by this template. In particular we obtain theorems of existence, uniqueness and non-existence for families of torus knots. We relate these families to resonant Hamiltonian bifurcations which occur as horseshoes are created in a one-parameter family of area preserving maps, and we also relate them to bifurcations of families of one-dimensional ‘quadratic like’ maps which can be studied by kneading theory. Thus, using knot theory, kneading theory and Hamiltonian bifurcation theory, we are able to connect a countable subsequence of “one-dimensional” bifurcations with a subsequence of “area-preserving” bifurcations in a two parameter family of suspensions in which horseshoes are created as the parameters vary. One implication is that infinitely many bifurcation sequences are reversed as one passes from the one dimensional to the area-preserving family: there are no universal routes to chaos!

Knotted periodic orbits in suspensions of Smale's horseshoe: Torus knots and bifurcation sequences

We investigate boundedness of the evolutione
 itH in the sense ofL
 2(ℝ3→L
 2(ℝ3) as well asL
 1(ℝ3→L
 ∞(ℝ3) for the non-selfadjoint operator
 
$$\mathcal{H} = \left[ \begin{gathered} - \Delta + \mu - V_1 \\ V_2 \\ \end{gathered} \right. \left. \begin{gathered} V_2 \\ \Delta - \mu + V_1 \\ \end{gathered} \right],$$

 where μ>0 andV
 1, V2 are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave, and the aforementioned bounds are needed in the study of nonlinear asymptotic stability of such standing waves. We derive our results under some natural spectral assumptions (corresponding to a ground state soliton of NLS), see A1)–A4) below, but without imposing any restrictions on the edges±μ of the essential spectrum. Our goal is to develop an “axiomatic approach,” which frees the linear theory from any nonlinear context in which it may have arisen.

/pdf/dispersive-estimates-for-schrodinger-operators-in-the-q68sjtvv4i.pdf

Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II

V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics. (The Mathematical Physics Monograph Series) IX + 286 S. m. Fig. New York/Amsterdam 1968. W. A. Benjamin, Inc. Preis geb. $ 14.75, brosch. $ 6.95 .

H.-G. Schöpf

Papers

V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics. (The Mathematical Physics Monograph Series) IX + 286 S. m. Fig. New York/Amsterdam 1968. W. A. Benjamin, Inc. Preis geb. $ 14.75, brosch. $ 6.95 .