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Beatrice Langella

Publications and source records attributed to Beatrice Langella.

16 recordsLinked to original sources
arXiv2026

Polynomial Prethermal Lifetimes in Non Smoothly Driven Quantum Systems

We study the dynamics of a quantum many-body lattice system with a local Hamiltonian subjected to a quasi-periodic driving with finite regularity. For sufficiently large driving frequencies, we prove that the system remains in a prethermal state for times growing polynomially with the frequency, and we show the optimality of this bound by constructing an explicit example that nearly saturates it. Within this prethermal regime, the dynamics is captured by an effective time-independent local Hamiltonian close to the undriven one. The proof relies on a non convergent normal form scheme, combined with original smoothing techniques for finitely differentiable local operators, and Lieb-Robinson bounds.

math-ph↗
arXiv2026

Transfer of energy for pure-gravity water waves with constant vorticity

We consider two-dimensional periodic gravity water waves with constant nonzero vorticity $γ$, in infinite depth and with periodic boundary conditions. We prove that, if the characteristic wave number $\frac{γ^2}{g}$ is rational, the system admits smooth small-amplitude solutions whose high Sobolev norms grow arbitrarily large while lower-order norms remain arbitrarily small, thereby exhibiting a genuine transfer of energy toward high frequencies. This yields the first rigorous construction of weakly turbulent solutions for a quasilinear hydrodynamic wave system, in a regime where the flow remains smooth. Moreover, the growth occurs simultaneously in the free surface and in the vertical component of the velocity at the interface, showing that the instability involves the full hydrodynamic evolution. The proof relies on a new mechanism for generating energy cascades in quasilinear dispersive PDEs with sublinear dispersion and a nonlinear transport structure. A central ingredient is to exploit quasi-resonances from 2-wave interactions to produce a transport operator that drives energy to high modes and causes Sobolev norm growth. A virial-type argument then shows that the resulting instability affects both the free surface elevation and the velocity field.

math.AP↗
arXiv2026

Nekhoroshev Theorem for time quasiperiodic perturbations of P-Steep systems

We prove a Nekhoroshev type result for a time quasiperiodic perturbation of an integrable Hamiltonian system. More precisely, we assume that the integrable part is analytic and fulfills a generic nondegeneracy condition introduced by Nekhoroshev and called P-Steepness. We add a small perturbation which depends in a quasiperiodic way on time (with Diophantine frequency) and prove that -- for times exponentially long with the inverse of the size $\varepsilon$ of the perturbation -- the actions of the unperturbed system remain approximately constant. The proof is based on an extension to the time dependent case of the proof {of classical Nekhoroshev's theorem} given by Guzzo, Chierchia and Benettin, which however requires new ideas in order to deal with the more complex geometry of resonances of the time dependent case.

math.DS↗
arXiv2025

Growth of Sobolev norms for completely resonant quantum harmonic oscillators on $\mathbb{R}^2$

We consider time dependently perturbed quantum harmonic oscillators in $\mathbb{R}^2$: $$ {\rm i} \partial_t u=\frac12(-\partial_{x_1}^2-\partial_{x_2}^2 + x_1^2+x_2^2)u +V(t, x, D)u, \qquad \ x \in \mathbb{R}^2, $$ where $V(t, x, D)$ is a selfadjoint pseudodifferential operator of degree zero, $2π$ periodic in time. We identify sufficient conditions on the principal symbol of the potential $V(t, x, D)$ that ensure existence of solutions exhibiting unbounded growth in time of their positive Sobolev norms and we show that the class of symbols satisfying such conditions is generic in the Fréchet space of classical $2π$- time periodic symbols of order zero. To prove our result we apply the abstract Theorem of arXiv:2101.09055v1 : the main difficulty is to find a conjugate operator $A$ for the resonant average of $V(t,x, D)$. We construct explicitly the symbol of the conjugate operator $A$, called escape function, combining techniques from microlocal analysis, dynamical systems and contact topology.

math.AP↗
arXiv2024

Growth of Sobolev norms in quasi integrable quantum systems

We prove an abstract result giving a $\langle t \rangle^\varepsilon$ upper bound on the growth of the Sobolev norms of a time-dependent Schrödinger equation of the form ${i} \dot ψ= H_0 ψ+ V (t)ψ$. Here $H_0$ 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. The proof is based on a quantum version of the proof of the classical Nekhoroshev theorem.

math.AP↗
arXiv2024

Prethermalization and conservation laws in quasi-periodically-driven quantum systems

We study conservation laws of a general class of quantum many-body systems subjected to an external time dependent quasi-periodic driving. {When the frequency of the driving is large enough or the strength of the driving is small enough, we prove a Nekhoroshev-type stability result: we show that the system exhibits a prethermal state for stretched exponentially long times in the perturbative parameter}. Moreover, we prove the quasi-conservation of the constants of motion of the unperturbed Hamiltonian and we analyze their physical meaning in examples of relevance to condensed matter and statistical physics.

math-ph↗
arXiv2024

Globally integrable quantum systems and their perturbations

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.

math.AP↗
arXiv2024

Almost global existence for some Hamiltonian PDEs on manifolds with globally integrable geodesic flow

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 Schrödinger 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.

math.AP↗
arXiv2023

Time periodic solutions of completely resonant Klein-Gordon equations on $\mathbb{S}^3$

We prove existence and multiplicity of Cantor families of small amplitude time periodic solutions of completely resonant Klein-Gordon equations on the sphere $\mathbb{S}^3$ with quadratic, cubic and quintic nonlinearity, regarded as toy models in General Relativity. The solutions are obtained by a variational Lyapunov- Schmidt decomposition, which reduces the problem to the search of mountain pass critical points of a restricted Euler-Lagrange action functional. Compactness properties of its gradient are obtained by Strichartz-type estimates for the solutions of the linear Klein-Gordon equation on $\mathbb{S}^3$.

math.AP↗
arXiv2022

Reducibility and nonlinear stability for a quasi-periodically forced NLS

Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schrödinger equation (NLS) on the two dimensional torus $\mathbb T^2:= (\mathbb R/2π\mathbb Z)^2$, we consider a quasi-periodically forced NLS equation on $\mathbb T^2$ arising from the linearization of the NLS at a KAM torus. We prove a reducibility result as well as long time stability of the origin. The main novelty is to obtain the precise asymptotic expansion of the frequencies which allows us to impose Melnikov conditions at arbitrary order.

math.AP↗
arXiv2021

Spectral asymptotics of all the eigenvalues of Schrödinger operators on flat tori

We study Schrödinger operators with Floquet boundary conditions on flat tori obtaining a spectral result giving an asymptotic expansion of all the eigenvalues. The expansion is in $λ^{-δ}$ with $δ\in(0,1)$ for most of the eigenvalues $λ$ (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.

math-ph↗
arXiv2020

A simple proof for a $C^\infty$ Nekhoroshev theorem

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.

math.DS↗
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↗