1,721,026 research outputs found
Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon
Full text linkIn this paper we consider linear, time dependent Schrödinger equations of the form i∂tψ=K0ψ+V(t)ψ, where K0 is a strictly positive selfadjoint operator with discrete spectrum and constant spectral gaps, and V(t) a smooth in time periodic potential. We give sufficient conditions on V(t) ensuring that K0+V(t) generates unbounded orbits. The main condition is that the resonant average of V(t), namely the average with respect to the flow of K0, has a nonempty absolutely continuous spectrum and fulfills a Mourre estimate. These conditions are stable under perturbations. The proof combines pseudodifferential normal form with dispersive estimates in the form of local energy decay. We apply our abstract construction to the Harmonic oscillator on R and to the half-wave equation on T; in each case, we provide large classes of potentials which are transporters
Generic Transporters for the Linear Time-Dependent Quantum Harmonic Oscillator on R
Full text linkIn this paper we consider the linear, time-dependent quantum Harmonic Schrdinger equation i partial derivative(t)u = 1/2(-partial derivative(x)(2) + x(2 ))u + V(t,x,d)u,x epsilon R, where v(t,x,D) is classical pseudodifferential operator of order 0, self-adjoint, and 2 pi periodic in time. We give sufficient conditions on the principal symbol of V(t,x,D) ensuring the existence of solutions displaying infinite time growth of Sobolev norms. These conditions are generic in the Frechet space of symbols. This shows that generic, classical pseudodifferential, 2 pi-periodic perturbations provoke unstable dynamics. The proof builds on the results of [36] and it is based on pseudodifferential normal form and local energy decay estimates. These last are proved exploiting Mourre's positive commutator theory
Tame majorant analyticity for the Birkhoff map of the defocusing nonlinear Schrödinger equation on the circle
Full text linkFor the defocusing nonlinear Schrö dinger equation on the circle, we construct a Birkhoff map Φ which is tame majorant analytic in a neighborhood of the origin. Roughly speaking, majorant analytic means that replacing the coefficients of the Taylor expansion of Φ by their absolute values gives rise to a series (the majorant map) which is uniformly and absolutely convergent, at least in a small neighborhood. Tame majorant analytic means that the majorant map of Φ fulfills tame estimates. The proof is based on a new tame version of the Kuksin-Perelman theorem (2010 Discrete Contin. Dyn. Syst. 1 1-24), which is an infinite dimensional Vey type theorem
Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators
Full text linkWe consider the semiclassical Schrödinger equation on Rd given by iħ∂tψ=(− [Formula presented] Δ+Wl(x))ψ+V(t,x)ψ, where Wl is an anharmonic trapping of the form Wl(x)= [Formula presented] ∑j=1dxj2l, l≥2 is an integer and ħ is a semiclassical small parameter. We construct a smooth potential V(t,x), bounded in time with its derivatives, and an initial datum such that the Sobolev norms of the solution grow at a logarithmic speed for all times of order log [Formula presented] (ħ−1). The proof relies on two ingredients: first we construct an unbounded solution to a forced mechanical anharmonic oscillator, then we exploit semiclassical approximation with coherent states to obtain growth of Sobolev norms for the quantum system which are valid for semiclassical time scales
Reducibility for a fast-driven linear Klein–Gordon equation
No full textWe prove a reducibility result for a linear Klein–Gordon equation with a quasi-periodic driving on a compact interval with Dirichlet boundary conditions. No assumptions are made on the size of the driving; however, we require it to be fast oscillating. In particular, provided that the external frequency is sufficiently large and chosen from a Cantor set of large measure, the original equation is conjugated to a time-independent, diagonal one. We achieve this result in two steps. First, we perform a preliminary transformation, adapted to fast oscillating systems, which moves the original equation in a perturbative setting. Then, we show that this new equation can be put to constant coefficients by applying a KAM reducibility scheme, whose convergence requires a new type of Melnikov conditions
Full description of Benjamin-Feir instability of stokes waves in deep water
Full text linkSmall-amplitude, traveling, space periodic solutions -called Stokes waves- of the 2 dimensional gravity water waves equations in deep water are linearly unstable with respect to long-wave perturbations, as predicted by Benjamin and Feir in 1967. We completely describe the behavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent is turned on. We prove in particular the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure "8", parameterized by the Floquet exponent, in full agreement with numerical simulations. Our new spectral approach to the Benjamin-Feir instability phenomenon uses a symplectic version of Kato's theory of similarity transformation to reduce the problem to determine the eigenvalues of a 4 x 4 complex Hamiltonian and reversible matrix. Applying a procedure inspired by KAM theory, we block-diagonalize such matrix into a pair of 2x2 Hamiltonian and reversible matrices, thus obtaining the full description of its eigenvalues
Benjamin-Feir instability of Stokes waves
No full textWe present the recent results in Berti et al. [Invent. Math. (2022), to appear] regarding the Benjamin-Feir instability of small amplitude Stokes waves in deep water. We completely describe the behavior of the four eigenvalues close to zero of the linearized water waves equations at the Stokes solution, as the Floquet exponent is turned on, proving the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure "8", in full agreement with numerical simulations
Local Well Posedness of the Euler-Korteweg Equations on T-d
No full textWe consider the Euler-Korteweg system with space periodic boundary conditions x is an element of T-d. We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity assumptions on the initial data
Strategies for the characterization and optimization of adsorptive stripping voltammetry with catalytic enhancement for ultratrace element determination: The case of iron 2,3-dihydroxynaphthalene complex with catalytic enhancement by atmospheric oxygen
No full textA comprehensive approach to the characterization and setup of metal determination by adsorptive stripping voltammetry with catalytic enhancement (CAdSV) is presented. The focus is on the understanding of the chemical features of these procedures to demonstrate which parameters can influence the analytical performances: the CAdSV method for the determination of iron at trace level using 2,3-dihydroxynaphthalene (DHN) is taken as case study. First, the ligand degradation was investigated by 1H NMR spectroscopy highlighting a significant degradation at alkaline pH of around 33% in 12 h. The use of degraded DHN had a detrimental effect on the analytical sensitivity, highlighting the need to frequently prepare the ligand. The thermodynamics of ligand and complex adsorption onto the working electrode (hanging mercury drop electrode, HMDE) was subsequently studied: both showed a strong adsorption onto the mercury surface (βDHN = 2.5·103 ± 5·102 L mol−1; βFe-DHN = 5.5·105 ± 8·104 L mol−1), but no competition for the mercury surface between the ligand and the complex was evident as determined by the multicomponent Langmuir isotherm. The mechanism of the electrode reaction was also investigated with and without the catalytic enhancement of the signal caused by air oxygen. The reduction of the complex Fe-DHN in purged solution showed α = 0.57 and k0 = 79 cm s−1, highlighting a quasireversible mechanism. The apparent catalytic constant (k'cat) was 168 s−1 for 1 mL sample volume: the simultaneous study of the kinetic and catalytic constant showed that the current signal was mostly influenced by the kinetic of the reaction
Long time dynamics of Schrödinger and wave equations on flat tori
No full textWe consider a class of linear time dependent Schrödinger equations and quasi-periodically forced nonlinear Hamiltonian wave/Klein Gordon and Schrödinger equations on arbitrary flat tori. For the linear Schrödinger equation, we prove a t ϵ (∀ϵ>0) upper bound for the growth of the Sobolev norms as the time goes to infinity. For the nonlinear Hamiltonian PDEs we construct families of time quasi-periodic solutions. Both results are based on “clusterization properties” of the eigenvalues of the Laplacian on a flat torus and on suitable “separation properties” of the singular sites of Schrödinger and wave operators, which are integers, in space–time Fourier lattice, close to a cone or a paraboloid. Thanks to these properties we are able to apply Delort abstract theorem [20] to control the speed of growth of the Sobolev norms, and Berti–Corsi–Procesi abstract Nash–Moser theorem [8] to construct quasi-periodic solutions
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