homogenization limits and wigner transforms we present a theory for carrying out homogenization limits for quadratic functions called energy densities of solutions of initial value problems ivps with anti self adjoint spatial pseudo differential operators pdos the approach is based on the introduction of phase space wigner matrix measures that are calculated by solving kinetic equations involving the spectral properties of the pdo the weak limits of the energy densities are then obtained by taking moments of the wigner measure the very general theory is illustrated by typical examples like semi classical limits of schrodinger equations with or without a periodic potential the homogenization limit of the acoustic equation in a periodic medium and the classical limit of the dirac equation 1997 john wiley sons inc

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice i homogeneous problems we present a computational approach for the wkb approximation of the wave function of an electron moving in a periodic one dimensional crystal lattice we derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the bloch spectrum of the schrodinger operator relying on the framework of the multibranch entropy solutions introduced by brenier and corrias we compute efficiently multiphase solutions using well adapted and simple numerical schemes in this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method

stability of banach spaces via nonlinear e isometries abstract in this paper we prove that the existence of an e isometry from a separable banach space x into y the james space or a reflexive space implies the existence of a linear isometry from x into y then we present a set valued mapping version lemma on non surjective e isometries of banach spaces using the above results we also discuss the rotundity and smoothness of banach spaces under the perturbation by e isometries

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

semiclassical and spectral analysis of oceanic waves in this work we prove that the shallow water flow subject to strong wind forcing and linearized around an adequate stationary profile develops for large times closed trajectories due to the propagation of rossby waves while poincare waves are shown to disperse the methods used in this paper involve semi classical analysis and dynamical systems for the study of rossby waves while some refined spectral analysis is required for the study of poincare waves due to the large time scale involved which is of diffractive type

non surjective coarse isometries between lp spaces abstract in this paper we study stability and weak stability of coarse isometries of banach spaces as a result we show that if a coarse isometry f l p 1 1 1 l p 2 2 2 1 p is weakly stable at each point of a schauder basis then there is a linear isometry u l p 1 1 1 l p 2 2 2 where j j j j 1 2 are finite measure spaces furthermore if f is uniformly weakly stable then f x u x o x when x as an application we obtain that p f x u x o x is equivalent to f x u x o x as x where p y u x is a projection with p 1

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 schrodinger 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

determinant bundles and virasoro algebras we consider the interplay of infinite dimensional lie algebras of virasoro type and moduli spaces of curves suggested by string theory we will see that the infinitesimal geometry of determinant bundles is governed by virasoro symmetries the mumford forms are just invariants of these symmetries the representations of virasoro algebra define twisted d modules on moduli spaces thesed modules are equations on correlators in conformal field theory

supercritical geometric optics for nonlinear schrodinger equations we consider the small time semi classical limit for nonlinear schrodinger equations with defocusing smooth nonlinearity for a super cubic nonlinearity the limiting system is not directly hyperbolic due to the presence of vacuum to overcome this issue we introduce new unknown functions which are defined nonlinearly in terms of the wave function itself this approach provides a local version of the modulated energy functional introduced by y brenier the system we obtain is hyperbolic symmetric and the justification of wkb analysis follows

a universal theorem for stability of e isometries of banach spaces abstract let x y be two banach spaces and f x y be a standard e isometry for some e 0 in this paper we show the following sharp weak stability inequality of f for every x x there exists y with x r such that x x f x 2 e r for all x x it is not only a sharp quantitative extension of figiel s theorem but it also unifies generalizes and improves a series of known results about stability of e isometries for example if the mapping f satisfies c f co f x f x y then it is equivalent to the following sharp stability theorem there is a linear surjective operator t y x of norm one such that t f x x 2 e for all x x when the e isometry f is surjective it is equivalent to omladic semrl s theorem there is a surjective linear isometry u x y so that f x u x 2 e for all x x

effective masses for laplacians on periodic graphs abstract we consider laplacians on both periodic discrete and periodic metric equilateral graphs their spectrum consists of an absolutely continuous part which is a union of non degenerate spectral bands and flat bands i e eigenvalues of infinite multiplicity we estimate effective masses associated with the ends of each spectral band in terms of geometric parameters of the graphs moreover in the case of the bottom of the spectrum we determine two sided estimates on the effective mass in terms of geometric parameters of the graphs the proof is based on floquet theory factorization of fiber operators perturbation theory and the relation between effective masses for laplacians on discrete and metric graphs obtained in our paper

semi classical asymptotics in solid state physics this article studies the schrodinger equation for an electron in a lattice of ions with an external magnetic field in a suitable physical scaling the ionic potential becomes rapidly oscillating and one can build asymptotic solutions for the limit of zero magnetic field by multiple scale methods from homogenization for the time dependent schrodinger equation this construction yields wave packets which follow the trajectories of the semiclassical model for the time independent equation one gets asymptotic eigenfunctions or quasimodes for the energy levels predicted by onsager s relation

on stability of nonlinear non surjective e isometries of banach spaces abstract let x y be two banach spaces e 0 and let f x y be an e isometry with f 0 0 in this paper we show first that for every x x there exists y with x r such that f x x x 4 e r for all x x making use of it we prove that if y is reflexive and if e y the annihilator of the subspace f y consisting of all functionals bounded on co f x f x is complemented in y then there is a bounded linear operator t y x with t such that t f x x 4 e for all x x if in addition y is gateaux smooth strictly convex and admitting the kadec klee property in particular locally uniformly convex then we have the following sharp estimate t f x x 2 e for all x x

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

homogenization in a periodic and time dependent potential this paper contains a study of the long time behavior of a diffusion process in a periodic potential the first goal is to determine a suitable rescaling of time and space so that the diffusion process converges to some homogeneous limit the issue of interest is to characterize the effective evolution equation the main result is that in some cases large drifts must be removed in order to get a diffusive asymptotic behavior this is applied to the homogenization of parabolic differential equations

a wigner measure analysis of the dufort frankel scheme for the schrodinger equation we apply wigner transform techniques to the analysis of the dufort frankel difference scheme for the schrodinger equation and to the continuous analogue of the scheme in the case of a small scaled planck constant semiclassical regime in this way we are able to obtain sharp conditions on the spatial temporal grid which guarantee convergence for average values of observables as the planck constant tends to zero the theory developed in this paper is not based on local and global error estimates and does not depend on whether or not caustics develop numerical test examples are presented to help interpret the theory and to compare the dufort frankel scheme to other difference schemes for the schr odinger equation

nonsurjective nearisometries of banach spaces abstract we obtain sharp approximation results for into nearisometries between l p spaces and nearisometries into a hilbert space our main theorem is the optimal approximation result for nearsurjective nearisometries between general banach spaces

effective mass theorems for nonlinear schrodinger equations we consider time dependent nonlinear schrodinger equations subject to smooth lattice periodic potentials plus additional confining potentials slowly varying on the lattice scale after an appropriate scaling we study the homogenization limit for vanishing lattice spacing assuming well prepared initial data the resulting effective dynamics is governed by a homogenized nonlinear schrodinger equation with an effective mass tensor depending on the initially chosen bloch eigenvalue the given results rigorously justify the use of the effective mass formalism for the description of bose einstein condensates on optical lattices

nonlinear waves in shallow honeycomb lattices the linear spectrum and corresponding bloch modes of shallow honeycomb lattices near dirac points are investigated via perturbation theory the dispersion relation is found to have threefold degeneracy at leading order with eigenvalue splitting at the following two orders i e the threefold eigenvalue splits into single and double values multiscale perturbation methods are employed to describe the nonlinear dynamics of the associated wave envelopes the dynamics of the envelope depends on different asymptotic balances whereupon a three level nonlinear dirac type equation or a two level nonlinear dirac equation is derived the analysis agrees well with direct numerical simulations

on the role of quadratic oscillations in nonlinear schrodinger equations we consider a nonlinear semi classical schrodinger equation for which it is known that quadratic oscillations lead to focusing at one point described by a nonlinear scattering operator if the initial data is an energy bounded sequence we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations the proof relies on a linearizability condition which can be expressed in terms of wigner measures when the initial data is a sum of such quadratic oscillations we prove that the associate solution is the superposition of the nonlinear evolution of each of them up to a small remainder term in an appendix we transpose those results to the case of the nonlinear schrodinger equation with harmonic potential

propagation of quantum expectations with husimi functions we analyze the dynamics of expectation values of quantum observables for the time dependent semiclassical schrodinger equation to benefit from the positivity of husimi functions we switch between observables obtained from weyl and anti wick quantization we develop and prove a second order egorov type propagation theorem with husimi functions by establishing transition and commutator rules for weyl and anti wick operators we provide a discretized version of our theorem and present numerical experiments for schrodinger equations in dimensions two and six that validate our results

wigner measures and observability for the schrodinger equation on the disk we analyse the structure of semiclassical and microlocal wigner measures for solutions to the linear schrodinger equation on the disk with dirichlet boundary conditions our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk we prove a structure theorem expressing the restriction of the wigner measures on each invariant torus in terms of second microlocal measures they are obtained by performing a finer localization in phase space around each of these tori at the limit of the uncertainty principle and are shown to propagate according to heisenberg equations on the circle our construction yields as corollaries a that the disintegration of the wigner measures is absolutely continuous in the angular variable which is an expression of the dispersive properties of the equation b an observability inequality saying that the l 2 norm of a solution on any open subset intersecting the boundary resp the l 2 norm of the neumann trace on any nonempty open set of the boundary controls its full l 2 norm resp h 1 norm these results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary

dispersion and strichartz estimates for the liouville equation we consider the liouville equation associated with a metric g of class c2 and we prove dispersion and strichartz estimates for the solution of this equation in terms of geodesics associated with g we introduce the notion of focusing and dispersive metric to characterize metrics such that the same dispersion estimate as in the euclidean case holds to deal with the case of non trapped long range perturbation of the euclidean metric we prove a global velocity moments effect on the solution in particular we obtain global in time strichartz estimates for metrics such that the dispersion estimate is not satisfied

opers and theta functions abstract we construct natural maps the klein and wirtinger maps from moduli spaces of semistable vector bundles over an algebraic curve x to affine spaces as quotients of the nonabelian theta linear series we prove a finiteness result for these maps over generalized kummer varieties moduli space of torus bundles leading us to conjecture that the maps are finite in general the conjecture provides canonical explicit coordinates on the moduli space the finiteness results give low dimensional parametrizations of jacobians in p 3g 3 for generic curves described by 2 functions or second logarithmic derivatives of theta we interpret the klein and wirtinger maps in terms of opers on x opers are generalizations of projective structures and can be considered as differential operators kernel functions or special bundles with connection the matrix opers analogues of opers for matrix differential operators combine the structures of flat vector bundle and projective connection and map to opers via generalized hitchin maps for vector bundles off the theta divisor the szego kernel gives a natural construction of matrix oper the wirtinger map from bundles off the theta divisor to the affine space of opers is then defined as the determinant of the szego kernel this generalizes the wirtinger projective connections associated to theta characteristics and the associated klein bidifferentials

stability of e isometric embeddings into banach spaces of continuous functions abstract let x be a banach space t be a compact hausdorff space and c t be the real banach space of all continuous functions on t endowed with the supremum norm we show that if there exists a standard e isometric embedding f x c t then there are nonempty closed subset s t and a linearly isometric embedding g x span f x s c s defined as g u lim n f 2 n u s 2 n for each u x satisfying that f u s g u 4 e for all u x making use of this result and the well known simultaneous extension operator e c s c t we also prove that the existence of a standard e isometric embedding f x c t implies the existence of a linearly isometric embedding e g x span e f x s c t whenever t is metrizable these conclusions generalize several well known results for any compact hausdorff space resp compact metric space t we further obtain that if g x is complemented in span f x s resp e g x is complemented in span e f x s then the standard e isometric embedding f x c t is stable

on perturbed metric preserved mappings and their stability characterizations abstract let x z be two real banach spaces and e 0 in this paper we first show that if there is a mapping f x z with f 0 0 satisfying f x f y x y e for all x y x then we can define a linear surjective isometry u x z n for some closed subspace n of z by an invariant mean of x we then show that the subspace n plays a crucial role in this paper for example 1 u n x is a w to w continuous surjective isometry and in particular if y span f x is reflexive then the mapping f is stable if and only if n is complemented in y 2 if y is reflexive and n is complemented in y then for any projection p y n the operator t u p satisfies t f x x 4 e for all x x and 3 if in addition y is gateaux smooth and locally uniformly convex then t u p satisfies the sharp estimate t f x x 2 e for all x x we present similar results for such mappings on general banach spaces

the phase flow method this paper introduces the phase flow method a novel accurate and fast approach for constructing phase maps for nonlinear autonomous ordinary differential equations the method operates by initially constructing the phase map for small times using a standard ode integration rule and builds up the phase map for larger times with the help of a local interpolation scheme together with the group property of the phase flow the computational complexity of building up the complete phase map is usually that of tracing a few rays in addition the phase flow method is provably and empirically very accurate once the phase map is available integrating the ode for initial conditions on the invariant manifold only makes use of local interpolation thus having constant complexity the paper develops applications in the field of high frequency wave propagation and shows how to use the phase flow method to 1 rapidly propagate wave fronts 2 rapidly calculate wave amplitudes along these wave fronts and 3 rapidly evaluate multiple wave arrival times at arbitrary locations

effective dynamics for bloch electrons peierls substitution and beyond we consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential x and vector potential a x with x d and 1 we prove that associated to an isolated family of bloch bands there exists an almost invariant subspace of l 2 d and an effective hamiltonian governing the evolution inside this subspace to all orders in to leading order the effective hamiltonian is given through the peierls substitution we explicitly compute the first order correction from a semiclassical analysis of this effective quantum hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics

homogenization of periodic systems with large potentials we consider the homogenization of a system of second order equations with a large potential in a periodic medium denoting by e the period the potential is scaled as e 2 under a generic assumption on the spectral properties of the associated cell problem we prove that the solution can be approximately factorized as the product of a fast oscillating cell eigenfunction and of a slowly varying solution of a scalar second order equation this result applies to various types of equations such as parabolic hyperbolic or eigenvalue problems as well as fourth order plate equation we also prove that for well prepared initial data concentrating at the bottom of a bloch band the resulting homogenized tensor depends on the chosen bloch band our method is based on a combination of classical homogenization techniques two scale convergence and suitable oscillating test functions and of bloch waves decomposition

interaction of modulated pulses in the nonlinear schrodinger equation with periodic potential abstract we consider a cubic nonlinear schrodinger equation with periodic potential in a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several bloch bands is studied the notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic interaction of these pulses

generalized stability of isometries abstract let x and y be real banach spaces and let e p 0 a mapping f x y is called an e p isometry if f x f y x y e x y p holds for all x y x a pair x y is p stable with respect to isometries if there exists a function 0 0 with lime 0 e 0 such that for every surjective e p isometry f x y there is a surjective isometry u x y satisfying the estimate f x u x e x p x x we show that every pair of banach spaces x y is p stable for 0 p r 2 r 2 is not 1 stable when p 1 a superstability phenomenon occurs for finite dimensional banach spaces

stability of almost surjective e isometries of banach spaces abstract we show that for every pair of banach spaces x and y and for every e isometry f x y with sup y s y lim inf t dist t y f x t 1 2 there exists an affine surjective isometry v y x such that f x v x 2 e for all x x

homogenization of periodic structures via bloch decomposition in this paper the classical problem of homogenization of elliptic operators in arbitrary domains with periodically oscillating coefficients is considered using bloch wave decomposition a new proof of convergence is furnished it sheds new light and offers an alternate way to view the classical results in a natural way this method leads us to work in the fourier space and thus in a framework dual to the one used by l tartar problemes d homogeneisation dans les equations aux derivees partielles cours peccot au college de france 1977 in his method of homogenization further this technique offers a nontraditional way of calculating the homogenized coefficients which is easy to implement in the computer