Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon

In 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

In 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

Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators

We 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

El trabajo teológico y docente del profesor Lucas F. Mateo-Seco

Este artículo pone de relieve la singular trayectoria intelectual y académica del Prof. Lucas F. Mateo Seco, analizando en primer lugar las constantes teológicas de su obra investigadora, en la que, sin duda, debe ser destacada su importante producción en torno al pensamiento de S. Gregorio de Nisa. Más tarde, tras hacer referencia a la obra manualística del Autor, cuyo vértice es su imponente tratado: “Dios Uno y Trino”, se pasa revista a los otros campos de trabajo cultivados con maestría por el teólogo Mateo-Seco: la teología y espiritualidad del sacerdocio, la enseñanza de S. Josemaría Escrivá, la teología de la liberación, etc

Full description of Benjamin-Feir instability of stokes waves in deep water

Small-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

Adiabatic Invariants for the FPUT and Toda Chain in the Thermodynamic Limit

We consider the Fermi\u2013Pasta\u2013Ulam\u2013Tsingou (FPUT) chain composed by N 6b 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature \u3b2- 1. Given a fixed 1 64 m 6a N, we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order \u3b2, for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics

Growth of Sobolev norms for abstract linear Schrodinger equations

We prove an abstract theorem giving a (t)ϵ bound (for all ϵ > 0) on the growth of the Sobolev norms in linear Schrodinger equations of the form i Ψ = H0ψ + V(t)ψ as t → ∞. The abstract theorem is applied to several cases, including the cases where (i) H0 is the Laplace operator on a Zoll manifold and V (t) a pseudodifferential operator of order smaller than 2; (ii) H0 is the (resonant or nonresonant) harmonic oscillator in Rd and V (t) a pseudodifferential operator of order smaller than that of H0 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 [MR17]

Physician's mistakes in the interpretation of spirometry

Background. The most recent ATS/ERS recommendations on lung function testing include a definition of airflow obstruction based on lower limit of normal (LLN) of FEV1/FVC and suggest to measure total lung capacity (TLC) in suspected cases of \u201cpseudo-restriction\u201d (normal FEV1/FVC ratio because of concomitant reductions in FEV1 and FVC), that can conceal airflow obstruction if the subject does not exhale long enough. Aims. To evaluate the skill of physicians in the interpretation of spirometry. Methods. A questionnaire focusing on the interpretation of five spirograms was administered to 127 physicians (aged 25-67yrs; 39% pulmonologists, 20% geriatrics). Correlates of spirometric misinterpretation were assessed by logistic regression. Results. Overall, 31% of physicians made at least one mistake in the interpretation of the spirograms administered. The percentage decreases to 15% among pulmonologists (OR=3.7; p=0.005). One quarter of physicians wrongly diagnosed airflow obstruction in a 75yrs old subject with FEV1/FVCLLN. About 1 out of 5 physicians did not recognize a mixed ventilatory defect (obstruction + restriction), while less than 15% (45% of pulmonologists) highlighted the need to measure TLC in suspected pseudo-restriction. Factors significantly associated with a lower amount of mistakes included higher n\ub0 of test performed, scientific articles read, respiratory congress attended, COPD and asthma patients visited in the last year. Conclusions. Inappropriate spirometric interpretation is not rare among physicians and airway obstruction is still frequently overdiagnosed among elderly. Diagnosis by pulmonologists and scientific update of physicians allow to reduce spirometric interpretative errors

Atmospheric fluctuations below 0.1 Hz during drift-scan solar diameter measurements

Measurements of the power spectrum of the seeing in the range 0.001-1 Hz have been performed in order to understand the criticity of the transits' method for solar diameter monitoring.Comment: 3 pages, 3 figures, proc. of the Fourth French-Chinese meeting on Solar Physics Understanding Solar Activity: Advances and Challenges, 15 - 18 November, 2011 Nice, Franc

Assessing mandibular body changes in growing subjects: a comparison of CBCT and reconstructed lateral cephalogram measurements

The aim of this study is to compare cone-beam computed tomography (CBCT) and bi-dimensional reconstructed lateral cephalograms (RLCs) in assessing mandibular body length and growth and to evaluate how mandibular reshaping influences the error in measuring mandibular body growth in bi-dimensional radiographs. Twenty-five patients with two CBCT scans taken at a mean distance of 2.21\u2009\ub1\u20090.5 years were selected. The following measurements were performed: right and left mandibular body length at each point in time, mandibular growth, inter-gonial distance and mandibular symphyseal angle. From each CBCT, an RLC was obtained, and mandibular body length and growth were measured. Data analysis revealed a statistically and clinically significant difference in CBCT and RLC regarding the mandibular length of each patient at each point in time. However, mandibular growth was almost identical. A linear regression was performed to predict growth distortion between RLCs and CBCT depending on the ratio between transverse and sagittal mandibular growth. The expected maximum and minimum distortion, however, appeared not to be significant. In fact, a second linear regression model and a Bland-Altman test revealed a strong correlation between measurements of average mandibular body growth by CBCT and RLCs. As the same distortion occurs in the first and second RLCs, bi-dimensional radiographs remain the method of choice in evaluating mandibular body growth