Exponential Stability in the Perturbed Central Force Problem

Abstract

We consider the spatial central force problem with a real analytic potential. We prove that for all potentials, but the Keplerian and the Harmonic ones, the Hamiltonian fulfills a nondegeneracy property needed for the applicability of Nekhoroshev's theorem. We deduce stability of the actions over exponentially long times when the system is subject to arbitrary analytic perturbation. The case where the central system is put in interaction with a slow system is also studied and stability over exponentially long time is proved.

Growth of Sobolev norms in quasi integrable quantum systems

“…We are finally going to verify steepness, by applying Theorem C.2 of the Appendix. Concerning this point, we remark that the paper [BFS18] contains the proof of the fact that in any central motion problem, but the Harmonic and the Keplerian ones, the condition of Theorem C.2 of the Appendix below is fulfilled (for the case of homogeneous potential this was already verified in [BF17]). We still have to verify A.iv.…”

“…In our application to the anharmonic oscillator the quantum actions are constructed by exploiting the theory of [CdV80,Cha83] in order to quantize the classical action variables. The steepness assumption is verified using the results of [Nie06,BF17,BFS18]. We remark that at present we are unable to deal with the 3 dimensional case, since the topology of the foliation of the classical integrable Hamiltonian system is very different in the 2 and in the 3 dimensional cases [BF16] and the results by Colin de Verdière [CdV80] do not apply to the 3-d case.…”

“…First (i) we recover the known results for tori [Bou99,Del10,BM19,BLM20a] and Zoll Manifolds [BGMR21], then (ii) we consider rotation invariant surfaces (following [CdV80,Del10]) and construct the operators A j by quantizing the classical action variables. The steepness of the classical Hamiltonian is then verified using tools from classical Hamiltonian theory ( [Nie06,BFS18]). The novelty of the result we get for rotation invariant surfaces is that we can deal with unbounded perturbations of the Laplacian.…”

On the Stable Eigenvalues of Perturbed Anharmonic Oscillators in Dimension Two

“…This is done using tools from degenerate KAM theory (see [Rüs01,BBM11]) and homogeneity of the frequency map. We also use some results from [FK04,BF17] (see also [BFS18]). The point (4) is solved using the same ideas developed by [Cha86].…”

Action-angle Variables for Generic 1D Mechanical Systems

“…)2 2 38 e s 0 ≥ 2 14 e s 0 η .Since 105 2 8 e s 0 η ≤ η ≤ r 4 /4 , we can apply Corollary 5.3 obtaining 106 sup E 2j−1 (r 4 /4,η )× Dr 0 By Cauchy estimate we get (noting that r 4 /8 ≥ η /2)sup E 2j−1 (r 4 /8,η /2)× Dr 0 ∂ EE I (2j−1)Now we note that if E satisfies (504), then it also satisfies (505) and, therefore,E ∈ E 2j−1 (r 4 /8, η /2) ,…”