Wigner Measure Propagation and Conical Singularity for General Initial Data

We study the evolution of Wigner measures of a family of solutions of a Schr\"odinger equation with a scalar potential displaying a conical singularity. Under a genericity assumption, classical trajectories exist and are unique, thus the question of the propagation of Wigner measures along these trajectories becomes relevant. We prove the propagation for general initial data.Comment: 24 pages, 1 figur

Propagation through Generic Level Crossings: A Surface Hopping Semigroup

We construct a surface hopping semigroup, which asymptotically describes nuclear propagation through crossings of electron energy levels. The underlying time-dependent Schrödinger equation has a matrix-valued potential, whose eigenvalue surfaces have a generic intersection of codimension two, three, or five in Hagedorn's classification. Using microlocal normal forms reminiscent of the Landau–Zener problem, we prove convergence to the true solution with an error of the order $\varepsilon^{1/8}$, where $\varepsilon$ is the semiclassical parameter. We present numerical experiments for an algorithmic realization of the semigroup illustrating the convergence of the algorithm

Wigner measures and codimension two crossings

This article gives a semiclassical description of nucleonic propagation through codimension two crossings of electronic energy levels. Codimension two crossings are the simplest energy level crossings, which affect the Born–Oppenheimer approximation in the zeroth order term. The model we study is a two-level Schrödinger equation with a Laplacian as kinetic operator and a matrix-valued linear potential, whose eigenvalues cross, if the two nucleonic coordinates equal zero. We discuss the case of well-localized initial data and obtain a description of the wavefunction’s two-scaled Wigner measure and of the weak limit of its position density, which is valid globally in time

Single switch surface hopping for molecular dynamics with transitions

A trajectory surface hopping algorithm is proposed, which stems from a mathematically rigorous analysis of propagation through conical intersections of potential energy surfaces. Since nonadiabatic transitions are only performed when a classical trajectory attains one of its local minimal surface gaps, the algorithm is called single switch surface hopping. Numerical experiments for a two mode Jahn–Teller system are presented, which illustrate the asymptotic justification of the method as well as its good performance in the physically relevant parameter range

Semiclassical approximations for Hamiltonians with operator-valued symbols

We consider the semiclassical limit of quantum systems with a Hamiltonian given by the Weyl quantization of an operator valued symbol. Systems composed of slow and fast degrees of freedom are of this form. Typically a small dimensionless parameter $\varepsilon\ll 1$ controls the separation of time scales and the limit $\varepsilon\to 0$ corresponds to an adiabatic limit, in which the slow and fast degrees of freedom decouple. At the same time $\varepsilon\to 0$ is the semiclassical limit for the slow degrees of freedom. In this paper we show that the $\varepsilon$-dependent classical flow for the slow degrees of freedom first discovered by Littlejohn and Flynn, coming from an \epsi-dependent classical Hamilton function and an $\varepsilon$-dependent symplectic form, has a concrete mathematical and physical meaning: Based on this flow we prove a formula for equilibrium expectations, an Egorov theorem and transport of Wigner functions, thereby approximating properties of the quantum system up to errors of order $\varepsilon^2$. In the context of Bloch electrons formal use of this classical system has triggered considerable progress in solid state physics. Hence we discuss in some detail the application of the general results to the Hofstadter model, which describes a two-dimensional gas of non-interacting electrons in a constant magnetic field in the tight-binding approximation.Comment: Final version to appear in Commun. Math. Phys. Results have been strengthened with only minor changes to the proofs. A section on the Hofstadter model as an application of the general theory was added and the previous section on other applications was remove

Ion and fluid transport properties of small airways in cystic fibrosis

Rationale: Small airways constitute amajor site of pathology in cystic fibrosis (CF) and provide most of the surface area of the conducting airways of the lung. Little is known, however, about the impact of CF on ion and fluid transport in small (bronchiolar) airways. Objectives: To describe the ion and fluid transport properties of CF bronchiolar epithelium. Methods: Primary cultures of human bronchial and bronchiolar (non-CF and CF) epithelial cells were obtained. The bioelectric properties were studied in Ussing chambers and the airway surface liquid (ASL) height was measured with confocal microscopy. Main Results: Primary cultures of ΔF508 CF bronchiolar epithelial cells displayed higher transepithelial resistance than non-CF cultures, whereas baseline short circuit current and amiloride-inhibitable short circuit current were similar in both preparations. The ASL height was significantly smaller in CF compared with non-CF preparations. In the presence of amiloride, addition of forskolin increased short circuit current in non-CF but not in CF bronchiolar cultures, and the ATP-induced increase in short circuit current was lower in CF than in non-CF cultures. Non-CF bronchiolar preparations displayed larger short circuit current and fluid secretion in responses to forskolin than non-CF bronchial preparations, suggesting that CFTR-dependent Cl- transport may play a more important role in the regulation of fluid transport in small airways than in large airways. Conclusion: In CF small airways, defective Cl- secretion combined with unregulated (persistent) Na+ absorption results in ASLdepletion

Nonlinear coherent states and Ehrenfest time for Schrodinger equation

We consider the propagation of wave packets for the nonlinear Schrodinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.Comment: 27 page

Evidence for validity and reliability of a french version of the FAAM

BACKGROUND: The Foot and Ankle Ability Measure (FAAM) is a self reported questionnaire for patients with foot and ankle disorders available in English, German, and Persian. This study plans to translate the FAAM from English to French (FAAM-F) and assess the validity and reliability of this new version.METHODS: The FAAM-F Activities of Daily Living (ADL) and sports subscales were completed by 105 French-speaking patients (average age 50.5 years) presenting various chronic foot and ankle disorders. Convergent and divergent validity was assessed by Pearson's correlation coefficients between the FAAM-F subscales and the SF-36 scales: Physical Functioning (PF), Physical Component Summary (PCS), Mental Health (MH) and Mental Component Summary (MCS). Internal consistency was calculated by Cronbach's Alpha (CA). To assess test re-test reliability, 22 patients filled out the questionnaire a second time to estimate minimal detectable changes (MDC) and intraclass correlation coefficients (ICC).RESULTS: Correlations for FAAM-F ADL subscale were 0.85 with PF, 0.81 with PCS, 0.26 with MH, 0.37 with MCS. Correlations for FAAM-F Sports subscale were 0.72 with PF, 0.72 with PCS, 0.21 with MH, 0.29 with MCS. CA estimates were 0.97 for both subscales. Respectively for the ADL and Sports subscales, ICC were 0.97 and 0.94, errors for a single measure were 8 and 10 points at 95% confidence and the MDC values at 95% confidence were 7 and 18 points.CONCLUSION: The FAAM-F is valid and reliable for the self-assessment of physical function in French-speaking patients with a wide range of chronic foot and ankle disorders

Disease Severity and Progression in Progressive Supranuclear Palsy and Multiple System Atrophy: Validation of the NNIPPS – PARKINSON PLUS SCALE

BACKGROUND The Natural History and Neuroprotection in Parkinson Plus Syndromes (NNIPPS) study was a large phase III randomized placebo-controlled trial of riluzole in Progressive Supranuclear Palsy (PSP, n = 362) and Multiple System Atrophy (MSA, n = 398). To assess disease severity and progression, we constructed and validated a new clinical rating scale as an ancillary study. METHODS AND FINDINGS Patients were assessed at entry and 6-montly for up to 3 years. Evaluation of the scale's psychometric properties included reliability (n = 116), validity (n = 760), and responsiveness (n = 642). Among the 85 items of the initial scale, factor analysis revealed 83 items contributing to 15 clinically relevant dimensions, including Activity of daily Living/Mobility, Axial bradykinesia, Limb bradykinesia, Rigidity, Oculomotor, Cerebellar, Bulbar/Pseudo-bulbar, Mental, Orthostatic, Urinary, Limb dystonia, Axial dystonia, Pyramidal, Myoclonus and Tremor. All but the Pyramidal dimension demonstrated good internal consistency (Cronbach α ≥ 0.70). Inter-rater reliability was high for the total score (Intra-class coefficient = 0.94) and 9 dimensions (Intra-class coefficient = 0.80-0.93), and moderate (Intra-class coefficient = 0.54-0.77) for 6. Correlations of the total score with other clinical measures of severity were good (rho ≥ 0.70). The total score was significantly and linearly related to survival (p<0.0001). Responsiveness expressed as the Standardized Response Mean was high for the total score slope of change (SRM = 1.10), though higher in PSP (SRM = 1.25) than in MSA (SRM = 1.0), indicating a more rapid progression of PSP. The slope of change was constant with increasing disease severity demonstrating good linearity of the scale throughout disease stages. Although MSA and PSP differed quantitatively on the total score at entry and on rate of progression, the relative contribution of clinical dimensions to overall severity and progression was similar. CONCLUSIONS The NNIPPS-PPS has suitable validity, is reliable and sensitive, and therefore is appropriate for use in clinical studies with PSP or MSA. TRIAL REGISTRATION ClinicalTrials.gov NCT00211224

Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves

In this paper, we model Rogue Waves as localized instabilities emerging from homogeneous and stationary background wavefields, under NLS dynamics. This is achieved in two steps: given any background Fourier spectrum P(k), we use the Wigner transform and Penrose’s method to recover spatially periodic unstable modes, which we call unstable Penrose modes. These can be seen as generalized Benjamin–Feir modes, and their parameters are obtained by resolving the Penrose condition, a system of nonlinear equations involving P(k). Moreover, we show how the superposition of unstable Penrose modes can result in the appearance of localized unstable modes. By interpreting the appearance of an unstable mode localized in an area not larger than a reference wavelength λ0 as the emergence of a Rogue Wave, a criterion for the emergence of Rogue Waves is formulated. Our methodology is applied to δ spectra, where the standard Benjamin–Feir instability is recovered, and to more general spectra. In that context, we present a scheme for the numerical resolution of the Penrose condition and estimate the sharpest possible localization of unstable modes. Keywords: Rogue Waves; Wigner equation; Nonlinear Schrodinger equation; Penrose modes; Penrose conditio