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In this paper we present the notion of globally integrable quantum system that we introduced in [BL22]: we motivate it using the spectral theory of pseudodifferential operators and then we give some results on linear and nonlinear perturbations of a globally integrable quantum system. In particular, we give a spectral result ensuring stability of most of its eigenvalues under relatively bounded perturbations, and two results controlling the growth of Sobolev norms when it is subject either to linear unbounded time dependent perturbations or a small nonlinear Hamiltonian nonlinear perturbation., Comment: 41 pages, 4 figures

We consider a modulated magnetic field, $B(t) = B_0 +\varepsilon f(\omega t)$, perpendicular to a fixed plane, where $B_0$ is constant, $\varepsilon>0$ and $f$ a periodic function on the torus ${\mathbb T}^n$. Our aim is to study classical and quantum dynamics for the corresponding Landau Hamiltonian. It turns out that the results depend strongly on the chosen gauge. For the Landau gauge the position observable is unbounded for "almost all" non resonant frequencies $\omega$. On the contrary, for the symmetric gauge we obtain that, for "almost all" non resonant frequencies $\omega$, the Landau Hamiltonian is reducible to a two dimensional harmonic oscillator and thus gives rise to bounded dynamics. The proofs use KAM algorithms for the classical dynamics. Quantum applications are given. In particular, the Floquet spectrum is absolutely continuous in the Landau gauge while it is discrete, of finite multiplicity, in symmetric gauge.

In this paper we prove an abstract result of almost global existence for small and smooth solutions of some semilinear PDEs on Riemannian manifolds with globally integrable geodesic flow. Some examples of such manifolds are Lie groups (including flat tori), homogeneous spaces and rotational invariant surfaces. As applications of the abstract result we prove almost global existence for a nonlinear Schr\"odinger equation with a convolution potential and for a nonlinear beam equation. We also prove $H^s$ stability of the ground state in NLS equation. The proof is based on a normal form procedure.

We prove a $C^\infty$ global version of Nekhoroshev theorem for time dependent Hamiltonians in $R^d\times T^d$. Precisely, we prove a result showing that for all times the actions of the unperturbed systems are bounded by a constant times $ \langle t\rangle^{\epsilon}$. We apply the result to the dynamics of a charged particle in $T^d$ subject to a time dependent electromagnetic field.

We consider a perturbation of the Benjamin Ono equation with periodic boundary conditions on a segment. We consider the case where the perturbation is Hamiltonian and the corresponding Hamiltonian vector field is analytic as a map form energy space to itself. Let $\epsilon$ be the size of the perturbation. We prove that for initial data close in energy norm to an $N$-gap state of the unperturbed equation all the actions of the Benjamin Ono equation remain $\cO(\epsilon^{\frac{1}{2(N+1)}})$ close to their initial value for times exponentially long with $\epsilon^{-\frac{1}{2(N+1)}}$., Comment: 22 pages

In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schr\"odinger equation with a convolution potential, a beam equation and a quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain's Lemma which provides a partition of the "resonant sites" of the Laplace operator on irrational tori.

We prove an abstract result giving a $\langle t\rangle^{\epsilon}$ upper bound on the growth of the Sobolev norms of a time dependent Schr\"odinger equation of the form $i \dot \psi = H_0 \psi + V(t) \psi $. $H_0$ is assumed to be the Hamiltonian of a steep quantum integrable system and to be a pseudodifferential operator of order ${\tt d}>1$; $V(t)$ is a time dependent family of pseudodifferential operators, unbounded, but of order ${\tt b}<{\tt d}$. The abstract theorem is then applied to perturbations of the quantum anharmonic oscillators in dimension 2 and to perturbations of the Laplacian on a manifold with integrable geodesic flow, and in particular Zoll manifolds, rotation invariant surfaces and Lie groups. We also cover the case of several particles on a Zoll manifold or on a Lie group, possibly obeying some restrictions due to the Fermionic or Bosonic nature of the particles. The proof is based a on quantum version of the proof of the classical Nekhoroshev theorem.

We prove a $\langle t\rangle^\varepsilon$ bound on the growth of Sobolev norms for unbounded time dependent perturbations of the Laplacian on flat tori.

We study Schr\"odinger operators with Floquet boundary conditions on flat tori obtaining a spectral result giving an asymptotic expansion of all the eigenvalues. The expansion is in $\lambda^{-\delta}$ with $\delta\in(0,1)$ for most of the eigenvalues $\lambda$ (stable eigenvalues), while it is a "directional expansion" for the remaining eigenvalues (unstable eigenvalues). The proof is based on a structure theorem which is a variant of the one proved in \cite{PS10,PS12} and on a new iterative quasimode argument., Comment: minor changes in the introduction. Some remarks added

We prove a $C^\infty$ version of the Nekhoroshev's estimate on the stability times of the actions in close to integrable Hamiltonian systems. The proof we give is a variant of the original Nekhoroshev's proof and it consists in first conjugating, globally in the phase space, and up to a small remainder, the system to a normal form. Then we perform the geometric part of the proof in the normalized variables. As a result, we obtain a proof which is simpler than the usual ones.

Consider the one-dimensional discrete Schr\"odinger operator $H_{\theta}$: $$(H_{\theta} q)_n=-(q_{n+1}+q_{n-1})+ V(\theta+n\omega) q_n \ , \quad n\in Z \ ,$$ with $\omega\in R^d$ Diophantine, and $V$ a real-analytic function on $ T^d=( R/2\pi Z)^d$. For $V$ sufficiently small, we prove the dispersive estimate: {for every $\phi\in\ell^1( Z)$,} $$ \| e^{-{\rm i}tH_{\theta}}\phi \|_{\ell^\infty} \leq K_0 \frac{ |\ln\varepsilon_0|^{a(\ln\ln(2+\langle t\rangle))^2 d}} {\langle t\rangle^{\frac13}} \|\phi\|_{\ell^1} \ , \quad \langle t \rangle:=\sqrt{1+t^2} \ ,$$ {with $a$ and $K_0$ two absolute constants} and $\varepsilon_0$ an analytic norm of $V$. The estimate holds for every $\theta\in T^d$.

We use a Hamiltonian normal form approach to study the dynamics of the water wave problem in the small amplitude long wave regime (KdV regime). If $\mu$ is the small parameter corresponding to the inverse of the wave length, we show that the normal form at order $\mu^5$ consists of two decoupled equation, one describing right going waves and the other describing left going waves. Performing a further non Hamiltonian transformation we conjugate each of these equations to a linear combination of the first three equations in the KdV hierarchy. At order $\mu^7$ we find nontrivial terms coupling the two counter-propagating waves., Comment: This second version contains an additional result estimating the error between solutions and an additional result of reduction to KDV5

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 Poincar\'e 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 Poincar\'e normal form and of the transformation to it, through a rapid convergence algorithm.

In this paper we study the spectrum of the operator \begin{equation} \label{ope} H:=(-\Delta)^{M/2}+\mathcal{V}\ , \quad M>0\ , \end{equation} on $L^2(\mathbb{R}^d/\Gamma)$, with $\Gamma$ 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} \lambda_\xi=|\xi|^M+Z(\xi)+O(\left|\xi\right|^{-\infty})\ , \end{equation} where $Z$ is a $C^\infty(\mathbb{R}^d)$ function (symbol) and $\xi\in\Gamma^*$ (the dual lattice of $\Gamma$).

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.

In this paper we study reducibility of time quasiperiodic perturbations of the quantum harmonic or anharmonic oscillator in one space dimension. We modify known algorithms obtaining a reducibility result which allows to deal with perturbations of order strictly larger than the ones considered in all the previous papers.

We prove that the spectrum of the discrete Schr\"odinger operator on $\ell^2(Z^2)$, $(\psi_{n,m})\mapsto -(\psi_{n+1,m} +\psi_{n-1,m} +\psi_{n,m+1} +\psi_{n,m-1})+V_n\psi_{n,m}$ is absolutely continuous.

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 $\beta^{2+\varsigma}$, $\beta$ being the inverse of the temperature and $\varsigma$ a positive number (we prove $\varsigma= 1/10$). The proof is obtained by adapting to the context of Gibbs measure for PDEs some tools of Hamiltonian perturbation theory.

We prove an abstract theorem giving a $\langle t\rangle^\epsilon$ bound ($\forall \epsilon>0$) on the growth of the Sobolev norms in linear Schr\"odinger equations of the form $i \dot \psi = H_0 \psi + V(t) \psi $ 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}.

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., Comment: 29 pages, 1 figure

We prove a reducibility result for a quantum harmonic oscillator in arbitrary dimensions with arbitrary frequencies perturbed by a linear operator which is a polynomial of degree two in $x_j$, $-i \partial_j$ with coefficients which depend quasiperiodically on time., Comment: fixed some misprints

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.

We study the Schr\"odinger equation on $\R$ with a potential behaving as $x^{2l}$ at infinity, $l\in[1,+\infty)$ and with a small time quasiperiodic perturbation. We prove that, if the perturbation belongs to a class of unbounded symbols including smooth potentials and magnetic type terms with controlled growth at infinity, then the system is reducible.

In this paper we study the dynamics of a soliton in the generalized NLS with a small external potential $\epsilon V$ 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 $\epsilon^{-r}$. 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 $\epsilon^{-r}$.

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 $R/N^\alpha$ (in discrete Sobolev-analytic norms) into a ball of radius $R'/N^\alpha$ (with $R,R'>0$ independent of $N$) if and only if $\alpha\geq2$. 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

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.

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 $\epsilon^{-1}$ 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 $ {\epsilon^{-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 $\epsilon^{-N}$ 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.

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 $(\omega^N_n)_{0 < n < N}$ 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, %$(2 \sin \frac{n\pi} {N})_{0 < n < N}$. 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 $\omega^N_n$ and $\omega^N_{N-n}$ at order $N^{-3}$ are given by a constant multiple of the n'th KdV frequencies $\omega^-_n$ and $\omega^+_n$ of two periodic potentials, $q_{-}$ respectively $q_+$, constructed in terms of the states considered. The frequencies $\omega^N_n$ 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.

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 $\pm (2-(2N)^{-2} \lambda ^\pm \_n + \cdots )$ where $(\lambda ^\pm \_n)\_{n \geq 0}$ 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 $L\_N$.

Consider an FPU chain composed of $N\gg 1$ particles, and endow the phase space with the Gibbs measure corresponding to a small temperature $\beta^{-1}$. Given a fixed $K0$) for initial data in a set of large measure. Furthermore, the time autocorrelation function of the energy of each packet does not decay significantly for times of order $\beta$. The restrictions on the shape of the packets are very mild. All estimates are uniform in the number $N$ of particles and thus hold in the thermodynamic limit $N\to\infty$, $\beta>0$.

We study the time of existence of the solutions of the following Schr\"odinger equation $$i\psi_t = (-\Delta)^s \psi +f(|\psi|^2)\psi, x \in \mathbb S^d, or x\in\T^d$$ where $(-\Delta)^s$ stands for the spectrally defined fractional Laplacian with $s>1/2$ and $f$ a smooth function. We prove an almost global existence result for almost all $s>1/2$.

We study the dynamics of an elastic body whose shape and position evolve due to the gravitational forces exerted by a pointlike planet. The main result is that, if all the deformations of the satellite dissipate some energy, then under a suitable nondegeneracy condition there are only three possible outcomes for the dynamics: (i) the orbit of the satellite is unbounded, (ii) the satellite falls on the planet, (iii) the satellite is captured in synchronous resonance i.e. its orbit is asymptotic to a motion in which the barycenter moves on a circular orbit, and the satellite moves rigidly, always showing the same face to the planet. The result is obtained by making use of LaSalle's invariance principle and by a careful kinematic analysis showing that energy stops dissipating only on synchronous orbits. We also use in quite an extensive way the fact that conservative elastodynamics is a Hamiltonian system invariant under the action of the rotation group.

We consider a Hamiltonian chain of weakly coupled anharmonic oscillators. It is well known that if the coupling is weak enough then the system admits families of periodic solutions exponentially localized in space (breathers). In this paper we prove asymptotic stability in energy space of such solutions. The proof is based on two steps: first we use canonical perturbation theory to put the system in a suitable normal form in a neighborhood of the breather, second we use dispersion in order to prove asymptotic stability. The main limitation of the result rests in the fact that the nonlinear part of the on site potential is required to have a zero of order 8 at the origin. From a technical point of view the theory differs from that developed for Hamiltonian PDEs due to the fact that the breather is not a relative equilibrium of the system.

In this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear Schr\"odinger equation on the real line. More precisely, we consider a symplectic time splitting integrator applied to a discrete nonlinear Schr\"odinger equation with additional Dirichlet boundary conditions on a large interval. We give conditions ensuring the existence of a numerical soliton which is close in energy norm to the continuous soliton. Such result is valid under a CFL condition between the time and space stepsizes. Furthermore we prove that if the initial datum is symmetric and close to the continuous soliton, then the associated numerical solution remains close to the orbit of the continuous soliton for very long times., Comment: 2012-25

We consider a ground state (soliton) of a Hamiltonian PDE. We prove that if the soliton is orbitally stable, then it is also asymptotically stable. The main assumptions are transversal nondegeneracy of the manifold of the ground states, linear dispersion (in the form of Strichartz estimates) and nonlinear Fermi Golden Rule. We allow the linearization of the equation at the soliton to have an arbitrary number of eigenvalues. The theory is tailor made for the application to the translational invariant NLS in space dimension 3. The proof is based on the extension of some tools of the theory of Hamiltonian systems (reduction theory, Darboux theorem, normal form) to the case of systems invariant under a symmetry group with unbounded generators., Comment: Some minor errors have been fixed

We consider the large number of particles limit of a periodic Toda lattice for a family of initial data close to the equilibrium state. We show that each of the two edges of the spectra of the corresponding Jacobi matrices is up to an error, determined by the spectra of two Hill operators, associated to this family. We then show that the spectra of the Jacobi matrices remain almost constant when the matrices evolve along the two limiting KdV equations. Finally we prove that the Toda actions, when appropriately renormalized, converge to the ones of KdV .

We consider the Gross-Petaevskii equation in 1 space dimension with a $n$-well trapping potential. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest n eigenvalues of the linear operator is slightly deformed by the nonlinear term into an almost invariant manifold M. Precisely, one has that solutions starting on M, or close to it, will remain close to M for times exponentially long with the inverse of the size of the nonlinearity. As heuristically expected the effective equation on M is a perturbation of a discrete nonlinear Schroedinger equation. We deduce that when the size of the nonlinearity is large enough then tunneling among the wells essentially disappears: that is for almost all solutions starting close to M their restriction to each of the wells has norm approximatively constant over the considered time scale. In the particular case of a double well potential we give a more precise result showing persistence or destruction of the beating motions over exponentially long times. The proof is based on canonical perturbation theory; surprisingly enough, due to the Gauge invariance of the system, no non-resonance condition is required.

We consider the time-dependent non linear Schrodinger equations with a double well potential in dimensions d =1 and d=2. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest two eigenvalues of the linear operator is almost invariant for any time.

This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on Zoll manifolds (e.g. spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on Zoll manifolds.