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Dario Bambusi

Publications and source records attributed to Dario Bambusi.

At least 37 records · Page 2Linked to original sources
arXiv2019

Exponential stability in the perturbed central force problem

We consider the spatial central force problem with a real analytic potential. We prove that for all analytic 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.

math-ph↗
arXiv2019

Convergence to normal forms of integrable PDEs

In an infinite dimensional Hilbert space we consider a family of commuting analytic vector fields vanishing at the origin and which are nonlinear perturbations of some fundamental linear vector fields. We prove that one can construct by the method of Poincaré normal form a local analytic coordinate transformation near the origin transforming the family into a normal form. The result applies to the KdV and NLS equations and to the Toda lattice with periodic boundary conditions. One gets existence of Birkhoff coordinates in a neighbourhood of the origin. The proof is obtained by directly estimating, in an iterative way, the terms of the Poincaré normal form and of the transformation to it, through a rapid convergence algorithm.

math.AP↗
arXiv2019

On the spectrum of the Schrödinger operator on $\mathbb{T}^d$: a normal form approach

In this paper we study the spectrum of the operator \begin{equation} \label{ope} H:=(-Δ)^{M/2}+\mathcal{V}\ , \quad M>0\ , \end{equation} on $L^2(\mathbb{R}^d/Γ)$, with $Γ$ a maximal dimension lattice in $\mathbb{R}^d$ and $\mathcal{V}$ a pseudodifferential operator of order strictly smaller than $M$. We prove that most of its eigenvalues admit the asymptotic expansion \begin{equation} \label{sim} λ_ξ=|ξ|^M+Z(ξ)+O(\left|ξ\right|^{-\infty})\ , \end{equation} where $Z$ is a $C^\infty(\mathbb{R}^d)$ function (symbol) and $ξ\inΓ^*$ (the dual lattice of $Γ$).

math-ph↗
arXiv2018

Reducibility of non-resonant transport equation on $T^d$ with unbounded perturbations

We prove reducibility of a transport equation on the $d$-dimensional torus $T^d$ with a time quasi-periodic unbounded perturbation. As far as we know this is the first example of a reducibility result for an equation in more than one dimensions with unbounded perturbations. Furthermore the unperturbed problem has eigenvalues whose differences are dense on the real axis.

math-ph↗
arXiv2018

A large probability averaging Theorem for the defocousing NLS

We consider the nonlinear Schroedinger equation on the one dimensional torus, with a defocousing polynomial nonlinearity and study the dynamics corresponding to initial data in a set of large measure with respect to the Gibbs measure. We prove that along the corresponding solutions the modulus of the Fourier coefficients is approximately constant for times of order $β^{2+ς}$, $β$ being the inverse of the temperature and $ς$ a positive number (we prove $ς= 1/10$). The proof is obtained by adapting to the context of Gibbs measure for PDEs some tools of Hamiltonian perturbation theory.

math-ph↗
arXiv2017

Growth of Sobolev norms for abstract linear Schrödinger Equations

We prove an abstract theorem giving a $\langle t\rangle^ε$ bound ($\forall ε>0$) on the growth of the Sobolev norms in linear Schrödinger equations of the form $i \dot ψ= H_0 ψ+ V(t) ψ$ when the time $t \to \infty$. The abstract theorem is applied to several cases, including the cases where (i) $H_0$ is the Laplace operator on a Zoll manifold and $V(t)$ a pseudodifferential operator of order smaller then 2; (ii) $H_0$ is the (resonant or nonresonant) Harmonic oscillator in $R^d$ and $V(t)$ a pseudodifferential operator of order smaller then $H_0$ depending in a quasiperiodic way on time. The proof is obtained by first conjugating the system to some normal form in which the perturbation is a smoothing operator and then applying the results of \cite{MaRo}.

math.AP↗
arXiv2016

Nekhoroshev theorem for perturbations of the central motion

In this paper we prove a Nekhoroshev type theorem for perturbations of Hamiltonians describing a particle subject to the force due to a central potential. Precisely, we prove that under an explicit condition on the potential, the Hamiltonian of the central motion is quasi-convex. Thus, when it is perturbed, two actions (the modulus of the total angular momentum and the action of the reduced radial system) are approximately conserved for times which are exponentially long with the inverse of the perturbation parameter.

math-ph↗
arXiv2016

Reducibility of 1-d Schroedinger equation with time quasiperiodic unbounded perturbations, I

We study the Schrödinger equation on $\R$ with a polynomial potential behaving as $x^{2l}$ at infinity, $1\leq l\in\N$ and with a small time quasiperiodic perturbation. We prove that if the symbol of the perturbation grows at most like $(ξ^2+x^{2l})^{β/(2l)}$, with $β<l+1$, then the system is reducible. Some extensions including cases with $β=2l$ are also proved. The result implies boundedness of Sobolev norms. The proof is based on pseudodifferential calculus and KAM theory.

math.DS↗
arXiv2015

From Toda to KdV

For periodic Toda chains with a large number $N$ of particles we consider states which are $N^{-2}$-close to the equilibrium and constructed by discretizing arbitrary given $C^2-$functions with mesh size $N^{-1}.$ Our aim is to describe the spectrum of the Jacobi matrices $L\_N$ appearing in the Lax pair formulation of the dynamics of these states as $N \to \infty$. To this end we construct two Hill operators $H\_\pm$ -- such operators come up in the Lax pair formulation of the Korteweg-de Vries equation -- and prove by methods of semiclassical analysis that the asymptotics as $N \rightarrow \infty $ of the eigenvalues at the edges of the spectrum of $L\_N$ are of the form $\pm (2-(2N)^{-2} λ^\pm \_n + \cdots )$ where $(λ^\pm \_n)\_{n \geq 0}$ are the eigenvalues of $H\_\pm $. In the bulk of the spectrum, the eigenvalues are $o(N^{-2})$-close to the ones of the equilibrium matrix. As an application we obtain asymptotics of a similar type of the discriminant, associated to $L\_N$.

math.AP↗
arXiv2015

Freezing of energy of a soliton in an external potential

In this paper we study the dynamics of a soliton in the generalized NLS with a small external potential $εV$ of Schwartz class. We prove that there exists an effective mechanical system describing the dynamics of the soliton and that, for any positive integer $r$, the energy of such a mechanical system is almost conserved up to times of order $ε^{-r}$. In the rotational invariant case we deduce that the true orbit of the soliton remains close to the mechanical one up to times of order $ε^{-r}$.

math-ph↗
arXiv2014

Birkhoff coordinates for the Toda Lattice in the limit of infinitely many particles with an application to FPU

In this paper we study the Birkhoff coordinates (Cartesian action angle coordinates) of the Toda lattice with periodic boundary condition in the limit where the number $N$ of the particles tends to infinity. We prove that the transformation introducing such coordinates maps analytically a complex ball of radius $R/N^α$ (in discrete Sobolev-analytic norms) into a ball of radius $R'/N^α$ (with $R,R'>0$ independent of $N$) if and only if $α\geq2$. Then we consider the problem of equipartition of energy in the spirit of Fermi-Pasta-Ulam. We deduce that corresponding to initial data of size $R/N^2$, $0<R\ll 1$, and with only the first Fourier mode excited, the energy remains forever in a packet of Fourier modes exponentially decreasing with the wave number. Finally we consider the original FPU model and prove that energy remains localized in a similar packet of Fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet. The proof of the theorem on Birkhoff coordinates is based on a new quantitative version of a Vey type theorem by Kuksin and Perelman which could be interesting in itself.

math-ph↗
arXiv2014

Some analytic results on the FPU paradox

We present some analytic results aiming at explaining the lack of thermalization observed by Fermi Pasta and Ulam in their celebrated numerical experiment. In particular we focus on results which persist as the number $N$ of particles tends to infinity. After recalling the FPU experiment and some classical heuristic ideas that have been used for its explanation, we concentrate on more recent rigorous results which are based on the use of (i) canonical perturbation theory and KdV equation, (ii) Toda lattice, (iii) a new approach based on the construction of functions which are adiabatic invariants with large probability in the Gibbs measure.

math-ph↗
arXiv2014

Normal Form and Energy Conservation of High Frequency Subsystems Without Nonresonance Conditions

We consider a system in which some high frequency harmonic oscillators are coupled with a slow system. We prove that up to very long times the energy of the high frequency system changes only by a small amount. The result we obtain is completely independent of the resonance relations among the frequencies of the fast system. More in detail, denote by $ε^{-1}$ the smallest high frequency. In the first part of the paper we apply the main result of [BG93] to prove almost conservation of the energy of the high frequency system over times exponentially long with $ {ε^{-1/n}} $ ($n$ being the number of fast oscillators). In the second part of the paper we give e new self-contained proof of a similar result which however is valid only over times of order $ε^{-N}$ with an arbitrary $N$. Such a second result is very similar to the main result of the paper [GHL13], which actually was the paper which stimulated our work.

math.DS↗
arXiv2013

Dynamics of periodic Toda chains with a large number of particles

For periodic Toda chains with a large number $N$ of particles we consider states which are $N^{-2}-$close to the equilibrium and constructed by discretizing any given $C^2-$functions with mesh size $N^{-1}$. For such states we derive asymptotic expansions of the Toda frequencies $(ω^N_n)_{0 < n < N}$ and the actions $(I^N_n)_{0 < n < N},$ both listed in the standard way, in powers of $N^{-1}$ as $N \to \infty$. %listed in accordance with the ordering of the frequencies at the equilibrium, %$(2 \sin \frac{nπ} {N})_{0 < n < N}$. At the two edges $n \sim 1$ and $N -n \sim 1$, the expansions of the frequencies are computed up to order $N^{-3}$ with an error term of higher order. Specifically, the coefficients of the expansions of $ω^N_n$ and $ω^N_{N-n}$ at order $N^{-3}$ are given by a constant multiple of the n'th KdV frequencies $ω^-_n$ and $ω^+_n$ of two periodic potentials, $q_{-}$ respectively $q_+$, constructed in terms of the states considered. The frequencies $ω^N_n$ for $n$ away from the edges are shown to be asymptotically close to the frequencies of the equilibrium. For the actions $(I^N_n)_{0 < n < N},$ asymptotics of a similar nature are derived.

math.AP↗