radiative transfer theory for high frequency power flows in fluid saturated poro visco elastic media some recent mathematical results are used to study the propagation features of the high frequency vibrational energy density in three dimensional fluid saturated isotropic poro visco elastic media the theory of high frequency asymptotics of the solutions of hyperbolic partial differential equations shows that their energy satisfies liouville type transport or radiative transfer equations for randomly heterogeneous materials for long propagation times these equations can be approached by diffusion equations the corresponding diffusion parameters mean free paths and diffusion constants associated with biot s linear model for such media are derived the analysis accounts for the thermal and viscous memory effects of the fluid phase and the viscous memory effect of the solid phase through time convolution operators in this respect it also extends the existing mathematical results

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

diffractive geometric optics for bloch wave packets we study for times of order 1 varepsilon solutions of wave equations which are fancyscript o varepsilon 2 modulations of an varepsilon periodic wave equation the solutions are of slowly varying amplitude type built on bloch plane waves with wavelength of order varepsilon we construct accurate approximate solutions of the three scale wkb type the leading profile is both transported at the group velocity and dispersed by a schrodinger equation given by the quadratic approximation of the bloch dispersion relation at the plane wave a ray average hypothesis of the small divisor type guarantees stability we introduce techniques related to those developed in nonlinear geometric optics which lead to new results even on time scales t fancyscript o 1 a pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems the leading term yields h1 asymptotics when the envelopes are only h1

on the mean field and classical limits of quantum mechanics the main result in this paper is a new inequality bearing on solutions of the n body linear schrodinger equation and of the mean field hartree equation this inequality implies that the mean field limit of the quantum mechanics of n identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation this result applies to the case of c1 1 interaction potentials the quantity measuring the approximation of the n body quantum dynamics by its mean field limit is analogous to the monge kantorovich or wasserstein distance with exponent 2 the inequality satisfied by this quantity is reminiscent of the work of dobrushin on the mean field limit in classical mechanics func anal appl 13 115 123 1979 our approach to this problem is based on a direct analysis of the n particle liouville equation and avoids using techniques based on the bbgky hierarchy or on second quantization

a field space based level set method for computing multi valued solutions to 1d euler poisson equations we present a field space based level set method for computing multi valued solutions to one dimensional euler poisson equations the system of these equations has many applications and in particular arises in semiclassical approximations of the schrodinger poisson equation the proposed approach involves an implicit eulerian formulation in an augmented space called field space which incorporates both velocity and electric fields into the configuration both velocity and electric fields are captured through common zeros of two level set functions which are governed by a field transport equation simultaneously we obtain a weighted density f by solving again the field transport equation but with initial density as starting data the averaged density is then resolved by the integration of the obtained f against the dirac delta function of two level set functions in the field space moreover we prove that such obtained averaged density is simply a linear superposition of all multi valued densities and the averaged field quantities are weighted superposition of corresponding multi valued ones computational results are presented and compared with some exact solutions which demonstrate the effectiveness of the proposed method

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

discrete hamilton jacobi theory we develop a discrete analogue of hamilton jacobi theory in the framework of discrete hamiltonian mechanics the resulting discrete hamilton jacobi equation is discrete only in time we describe a discrete analogue of jacobi s solution and also prove a discrete version of the geometric hamilton jacobi theorem the theory applied to discrete linear hamiltonian systems yields the discrete riccati equation as a special case of the discrete hamilton jacobi equation we also apply the theory to discrete optimal control problems and recover some well known results such as the bellman equation discrete time hjb equation of dynamic programming and its relation to the costate variable in the pontryagin maximum principle this relationship between the discrete hamilton jacobi equation and bellman equation is exploited to derive a generalized form of the bellman equation that has controls at internal stages

a hybrid phase flow method for hamiltonian systems with discontinuous hamiltonians in this paper we propose a new phase flow method for hamiltonian systems with discontinuous hamiltonians in the original phase flow method introduced by ying and candes j comput phys 220 2006 pp 184 215 the phase map should be smooth to ensure the accuracy of the interpolation such an interpolation is inaccurate if the phase map is nonsmooth for example when the hamiltonian is discontinuous we modify the phase flow method using a discontinuous hamiltonian solver and establish the stability for piecewise constant potentials of such a solver this extends the applicability of the highly efficient phase flow method to singular hamiltonian systems with a mild increase of algorithm complexity such a particle method can be useful for the computation of high frequency waves through interfaces

a posteriori error control and adaptivity for crank nicolson finite element approximations for the linear schrodinger equation we derive optimal order a posteriori error estimates for fully discrete approximations of linear schrodinger type equations in the l infty l 2 l l 2 norm for the discretization in time we use the crank nicolson method while for the space discretization we use finite element spaces that are allowed to change in time the derivation of the estimators is based on a novel elliptic reconstruction that leads to estimates which reflect the physical properties of schrodinger equations the final estimates are obtained using energy techniques and residual type estimators various numerical experiments for the one dimensional linear schrodinger equation in the semiclassical regime verify and complement our theoretical results the numerical implementations are performed with both uniform partitions and adaptivity in time and space for adaptivity we further develop and analyze an existing time space adaptive algorithm to the cases of schrodinger equations the adaptive algorithm reduces the computational cost substantially and provides efficient error control for the solution and the observables of the problem especially for small values of the planck constant

numerical microlocal analysis of harmonic wavefields we present and test a numerical method which given an analytical or numerical solution of the helmholtz equation in a neighborhood of a fixed observation point and assuming that the geometrical optics approximation is relevant determines at this point the number of crossing rays and computes their directions and associated complex amplitudes

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

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

macroscopic behavior of microscopic oscillations in harmonic lattices via wigner husimi transforms we consider the dynamics of infinite harmonic lattices in the limit of the lattice distance tending to 0 we allow for general polyatomic crystals but assume exact periodicity such that the system can be solved in principle by fourier transform and linear algebra methods

transport equations for a general class of evolution equations with random perturbations we derive transport equations from a general class of equations of form iut h x d u v x d u where h x d and v x d are pseudodifferential operators weyl operator with symbols h x k and v x k where h x k being polynomial in k and smooth in x v x k is a mean zero random function and is stationary in space variable we also consider system of equations in the above form such equations cover many of the equations that arise in wave propagations such as those considered in a paper by ryzhik papanicolaou and keller wave motion 24 327 370 1996 our results generalize those by ryzhik papanicolau and keller

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice ii impurities confinement and bloch oscillations we present a computational approach for the wkb approximation of the wavefunction of an electron moving in a periodic one dimensional crystal lattice by means of a nonstrictly hyperbolic system whose flux function stems from the bloch spectrum of the schrodinger operator this second part focuses on the handling of the source terms which originate from adding a slowly varying exterior potential physically relevant examples are the occurrence of bloch oscillations in case it is linear a quadratic one modelling a confining field and the harmonic coulomb term resulting from the inclusion of a donor impurity inside an otherwise perfectly homogeneous lattice

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 hybrid phase flow method for solving the liouville equation in a bounded domain the phase flow method originally introduced in l x ying and e j candes j comput phys 220 2006 pp 184 215 can efficiently solve autonomous ordinary differential equations in s jin h wu and z y huang siam j sci comput 31 2008 pp 1303 1321 the method was generalized to solve hamiltonian system where the hamiltonian function was discontinuous however both of these methods require a phase flow map constructed on an invariant manifold this can increase computational cost when the invariant domain is big or unbounded following the idea of s jin h wu and z y huang siam j sci comput 31 2008 pp 1303 1321 we propose a hybrid phase flow method for solving the liouville equation in a bounded domain which is smaller than the invariant manifold of a phase flow map by using some proper boundary conditions this method can help solve the problem where the invariant manifold of a phase flow map determined by the liouville equation is unbounded we verify numerical accuracy and efficiency by several examples of the semiclassical limit of the schrodinger equation analysis of numerical stability and convergence is given for the semiclassical limit equation with inflow boundary condition

boundary triples and weyl m functions for powers of the jacobi differential operator abstract the abstract theory of boundary triples is applied to the classical jacobi differential operator and its powers in order to obtain the weyl m function for several self adjoint extensions with interesting boundary conditions separated periodic and those that yield the friedrichs extension these matrix valued nevanlinna herglotz m functions are to the best knowledge of the author the first explicit examples to stem from singular higher order differential equations the creation of the boundary triples involves taking pieces determined in 26 of the principal and non principal solutions of the differential equation and putting them into the sesquilinear form to yield maps from the maximal domain to the boundary space these maps act like quasi derivatives which are usually not well defined for all functions in the maximal domain of singular expressions however well defined regularizations of quasi derivatives are produced by putting the pieces of the non principal solutions through a modified gram schmidt process

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

self averaging of wigner transforms in random media we establish the self averaging properties of the wigner transform of a mixture of states in the regime when the correlation length of the random medium is much longer than the wave length but much shorter than the propagation distance the main ingredients in the proof are the error estimates for the semiclassical approximation of the wigner transform by the solution of the liouville equations and the limit theorem for two particle motion along the characteristics of the liouville equations the results are applied to a mathematical model of the time reversal experiments for the acoustic waves and self averaging properties of the re transmitted wave are proved

propagation observation and control of waves approximated by finite difference methods this paper surveys several topics related to the observation and control of wave propagation phenomena modeled by finite difference methods the main focus is on the property of observability corresponding to the question of whether the total energy of solutions can be estimated from partial measurements on a subregion of the domain or boundary the mathematically equivalent property of controllability corresponds to the question of whether wave propagation behavior can be controlled using forcing terms on that subregion as is often desired in engineering applications observability controllability of the continuous wave equation is well understood for the scalar linear constant coefficient case that is the focus of this paper however when the wave equation is discretized by finite difference methods the control for the discretized model does not necessarily yield a good approximation to the control for the original continuous problem in other words the classical convergence consistency stability property of a numerical scheme does not suffice to guarantee its suitability for providing good approximations to the controls that might be needed in applications observability controllability may be lost under numerical discretization as the mesh size tends to zero due to the existence of high frequency spurious solutions for which the group velocity vanishes this phenomenon is analyzed and several remedies are suggested including filtering tychonoff regularization multigrid methods and mixed finite element methods we also briefly discuss these issues for the heat beam and schrodinger equations to illustrate that diffusive and dispersive effects may help to retain the observability controllability properties at the discrete level we conclude with a list of open problems and future subjects for research

self averaging scaling limits for random parabolic waves we consider several types of scaling limits for the wigner moyal equation of the parabolic waves in random media the limiting cases of which include the standard radiative transfer limit the geometrical optics limit and the white noise limit we show under fairly general assumptions on the random refractive index field that sufficient amount of medium diversity thus excluding the white noise limit leads to statistical stability or self averaging in the sense that the limiting law is deterministic and is governed by one of the 6 different types of transport boltzmann or fokker planck equations depending on the specific scaling involved we discuss the connection to the statistical stability of time reversal procedure and the decoherence effect in quantum mechanics

asymptotics of the solutions of the random schrodinger equation we consider solutions of the schrodinger equation with a weak time dependent random potential it is shown that when the two point correlation function of the potential is rapidly decaying then the fourier transform hat zeta epsilon t xi of the appropriately scaled solution converges point wise in to a stochastic complex gaussian limit on the other hand when the two point correlation function decays slowly we show that the limit of hat zeta epsilon t xi has the form hat zeta 0 xi rm exp ib kappa t xi where b t is a fractional brownian motion

nonlinear schrodinger equation with a white noise potential phase space approach to spread and singularity we propose a phase space formulation for the nonlinear schr odinger equation with a white noise potential in order to shed light on two issues the rate of spread and the singularity formation in the average sense our main tools are the energy law and the variance identity the method is completely elementary for the problem of wave spread we show that the ensemble averaged dispersion in the critical or defocusing case follows the cubic in time law while in the supercritical and subcritical focusing cases the cubic law becomes an upper and lower bound respectively we have also found that in the critical and supercritical focusing cases the presence of a white noise random potential results in different conditions for singularity with positive probability from the homogeneous case but does not prevent singularity formation we show that in the supercritical focusing case the ensemble averaged self interaction energy and the momentum variance can exceed any fixed level in a finite time with positive probability c 2005 elsevier b v all rights reserved

wkb analysis of bohmian dynamics we consider a semiclassically scaled schrodinger equation with wkb initial data we prove that in the classical limit the corresponding bohmian trajectories converge locally in measure to the classical trajectories before the appearance of the first caustic in a second step we show that after caustic onset this convergence in general no longer holds in addition we provide numerical simulations of the bohmian trajectories in the semiclassical regime that illustrate the above results 2014 wiley periodicals inc

a uniformly accurate ua multiscale time integrator pseudospectral method for the dirac equation in the nonrelativistic limit regime we propose and rigourously analyze a multiscale time integrator fourier pseudospectral mti fp method for the dirac equation with a dimensionless parameter varepsilon in 0 1 which is inversely proportional to the speed of light in the nonrelativistic limit regime i e 0 varepsilon ll 1 the solution exhibits highly oscillatory propagating waves with wavelength o varepsilon 2 and o 1 in time and space respectively due to the rapid temporal oscillation it is quite challenging in designing and analyzing numerical methods with uniform error bounds in varepsilon in 0 1 we present the mti fp method based on properly adopting a multiscale decomposition of the solution of the dirac equation and applying the exponential wave integrator with appropriate numerical quadratures by a careful study of the error propagation and using the energy method we establish two independent error estimates via two different mathematical approaches as h m 0 frac tau 2 varepsilon 2 and h m 0 tau 2 varepsilon 2 where h is the mesh size tau is the time step and m 0 depends on the regularity of the solution these two error bounds immediately imply that the mti fp method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau for all varepsilon in 0 1 and optimally with quadratic convergence rate at o tau 2 in the regimes when either varepsilon o 1 or 0 varepsilon lesssim tau numerical results are reported to demonstrate that our error estimates are optimal and sharp finally the mti fp method is applied to study numerically the convergence rates of the solution of the dirac equation to those of its limiting models when varepsilon to0

optimal control of partitioned hybrid systems via discrete time hamilton jacobi theory abstract computational framework for optimal control of hybrid systems with a partitioned state space is presented it is shown that necessary conditions for optimality for a discrete time dynamic system can be solved concurrently for various boundary conditions according to the recent development of discrete time hamilton jacobi theory this unique property is utilized to construct computationally efficient numerical optimization of hybrid systems where discrete switching dynamics occurs at the boundary between partitions of the configuration space a benchmark example shows that the proposed approach has substantial computational advantages compared with the existing ones

a particle method for the semiclassical limit of the schrodinger equation and the vlasov poisson equations when using standard deterministic particle methods point values of the computed solutions have to be recovered from their singular particle approximations by using some smoothing procedure the choice of the smoothing procedure is rather flexible moreover there is always a parameter associated with the smoothing procedure if this parameter is too large the numerical solution loses its accuracy if it is too small oscillation appears no explicit formula has been given on how to choose this parameter in this paper we develop a particle method for the semiclassical limit of the schrodinger equation and the vlasov poisson equations in which we use the property of conservation of charge which was studied in d wei kinetic and related models 3 2010 pp 729 754 to construct the density this method avoids the recovery step of the particle methods thus it is simpler and more accurate in particular it gives more accurate field quantities consequently we apply this method to the vlasov poi

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

homogenization of the schrodinger equation and effective mass theorems we study the homogenization of a schrodinger equation with a large periodic potential denoting by the period the potential is scaled as 2 we obtain a rigorous derivation of so called effective mass theorems in solid state physics more precisely for well prepared initial data concentrating on a bloch eigenfunction we prove that the solution is approximately the product of a fast oscillating bloch eigenfunction and of a slowly varying solution of an homogenized schrodinger equation the homogenized coefficients depend on the chosen bloch eigenvalue and the homogenized solution may experience a large drift the homogenized limit may be a system of equations having dimension equal to the multiplicity of the bloch eigenvalue our method is based on a combination of classical homogenization techniques two scale convergence and suitable oscillating test functions and of bloch waves decomposition

white noise and geometrical optics limits of wigner moyal equation for wave beams in turbulent media starting with the wigner distribution formulation for beam wave propagation in holder continuous non gaussian random refractive index fields we show that the wave beam regime naturally leads to the white noise scaling limit and converges to a gaussian white noise model which is characterized by the martingale problem associated to a stochastic differential integral equation of the ito type in the simultaneous geometrical optics the convergence to the gaussian white noise model for the liouville equation is also established if the ultraviolet cutoff or the fresnel number vanishes sufficiently slowly the advantage of the gaussian white noise model is that its n point correlation functions are governed by closed form equations

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

discrete variational hamiltonian mechanics the main contribution of this paper is to present a canonical choice of a hamiltonian theory corresponding to the theory of discrete lagrangian mechanics we make use of lagrange duality and follow a path parallel to that used for construction of the pontryagin principle in optimal control theory we use duality results regarding sensitivity and separability to show the relationship between generating functions and symplectic integrators we also discuss connections to optimal control theory and numerical algorithms

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

kinetic limit for wave propagation in a random medium we study crystal dynamics in the harmonic approximation the atomic masses are weakly disordered in the sense that their deviation from uniformity is of the order e the dispersion relation is assumed to be a morse function and to suppress crossed recollisions we then prove that in the limit e 0 the disorder averaged wigner function on the kinetic scale time and space of order e 1 is governed by a linear boltzmann equation

relative form boundedness and compactness for a second order differential operator if a and l are self adjoint operators on a hilbert space h such that a is nonnegative and l ei for some e 0 we study conditions under which a is form bounded or form compact with respect to l and contrast these concepts with the stronger properties that a be relatively operator bounded or compact with respect to l in particular several definitions of form compactness are shown to be equivalent the principal application of the theory is to self adjoint second order operators in this case conditions that a be form bounded or form compact with respect to l are shown in some cases to be necessary and sufficient examples include the energy operator of the hydrogen atom

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

numerical approximation of quadratic observables of schrodinger type equations in the semi classical limit we apply wigner transform techniques to the analysis of difference methods for schrodinger type equations in the case of a small planck constant 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 caustics develop or not numerical test examples are presented to help interpret the theory

global solution for the optimal feedback control of the underactuated heisenberg system we present a global solution for an optimal feedback controller of the underactuated heisenberg system or nonholonomic integrator employing a recently developed technique based on generating functions appearing in the hamilton jacobi theory we circumvent a singularity caused by under actuation to develop a nonlinear optimal feedback control in an implicitly analytical form the systematic procedure to deal with under actuation indicates that generating functions should be effective tools for solving general underactuated optimal control problems

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

propagation through conical crossings an asymptotic semigroup we consider the standard model problem for a conical intersection of electronic surfaces in molecular dynamics our main result is the construction of a semi group in order to approximate the wigner function associated with the solution of the schrodinger equation at leading order in the semiclassical parameter the semigroup stems from an underlying markov process that combines deterministic transport along classical trajectories within the electronic surfaces and random jumps between the surfaces near the crossing our semigroup can be viewed as a rigorous mathematical counterpart of so called trajectory surface hopping algorithms which are of major importance in molecular simulations in chemical physics the key point of our analysis the incorporation of the nonadiabatic transitions is based on the landau zener type formula of fermanian kammerer and gerard10 for the propagation of two scale wigner measures through conical crossings 2005 wiley periodicals inc

elastic wave propagation in a 3 d unbounded random heterogeneous medium coupled with a bounded medium application to seismic soil structure interaction sssi we present a sub structuring method for the coupling between a large elastic structure and a stratified soil half space exhibiting random heterogeneities over a bounded domain and impinged by incident waves both media are also weakly dissipative the concept of interfaces classically used in sub structuring methods is extended to volume interfaces in the proposed approach the random dimension of the stochastic fields modelling the heterogeneities in the soil is reduced by introducing a karhunen loeve expansion of these stochastic fields the coupled overall problem is solved by monte carlo simulation techniques a realistic example of a large industrial structure interacting with an uncertain stratified soil medium under earthquake is finally presented this case study and others validate the presented methodology and its ability to handle complex mechanical systems copyright 2002 john wiley sons ltd

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice iii from ab initio models to wkb for schrodinger poisson this work is concerned with the semiclassical approximation of the schrodinger poisson equation modeling ballistic transport in a 1d periodic potential by means of wkb techniques it is derived by considering the mean field limit of a n body quantum problem then k multivalued solutions are adapted to the treatment of this weakly nonlinear system obtained after homogenization without taking into account for pauli s exclusion principle numerical experiments display the behaviour of self consistent wave packets and screening effects

the effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory this study of the effect of variable change on differential operators was motivated by recent papers of tipler 131 and hinton and lewis 37 which appeared in the same issue of the journal of d erential equations tipler gave a geometric proof 13 p 1671 of the following result of hawking and penrose see 29 301 let f t be a real valued continuous nonnegative function on w co such that f b 0 for some point b then there exists a pair of conjugate points on co co for the differential equation