global existence for the einstein vacuum equations in wave coordinates we prove global stability of minkowski space for the einstein vacuum equations in harmonic wave coordinate gauge for the set of restricted data coinciding with the schwarzschild solution in the neighborhood of space like infinity the result contradicts previous beliefs that wave coordinates are unstable in the large and provides an alternative approach to the stability problem originally solved for unrestricted data in a different gauge and with a precise description of the asymptotic behavior at null infinity by d christodoulou and s klainerman

strichartz estimates for the wave and schrodinger equations with the inverse square potential abstract we prove spacetime weighted l 2 estimates for the schrodinger and wave equation with an inverse square potential we then deduce strichartz estimates for these equations

convergence acceleration for boundary value problems with singularities using the e algorithm in the present work we use the e algorithm to accelerate the convergence of finite difference schemes for ordinary differential equations as a model a boundary value problem for a second order differential equation is considered where the right hand side may have different kinds of singularities an asymptotic error expansion is obtained enabling the use of the e algorithm to accelerate the convergence the applicability and the efficiency of the e algorithm are discussed and illustrated by numerical examples

convergence analysis of two stage waveform relaxation method for the initial value problems we present a class of two stage waveform relaxation methods for solving the initial value problems of ordinary differential equations by making use of the laplace transform we discuss the convergence of these methods and derive sufficient conditions for guaranteeing their convergence when the system matrices are specifically h matrices also we discuss the monotonicity of the convergence sequence further we compare the sequences generated by two different inner splittings of the same initial value problem

the quantum n body problem this selective review is written as an introduction to the mathematical theory of the schrodinger equation for n particles characteristic for these systems are the cluster properties of the potential in configuration space which are expressed in a simple geometric language the methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of a number of basic and by now classical results the central theme is the interplay between the spectral theory of n body hamiltonians and the space time and phase space analysis of bound states and scattering states

a proof of price s law for the collapse of a self gravitating scalar field a well known open problem in general relativity dating back to 1972 has been to prove price s law for an appropriate model of gravitational collapse this law postulates inverse power decay rates for the gravitational radiation flux through the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates it is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon in this paper we prove a well defined upper bound formulation of price s law for the collapse of a self gravitating scalar field with spherically symmetric initial data we also allow the presence of an additional gravitationally coupled maxwell field our results are obtained by a new mathematical technique for understanding the long time behavior of large data solutions to the resulting coupled non linear hyperbolic system of p d e s in 2 independent variables the technique is based on the interaction of the conformal geometry the celebrated red shift effect and local energy conservation we feel it may be relevant for the problem of non linear stability of the kerr solution when combined with previous work of the first author concerning the internal structure of charged black holes which had assumed the validity of price s law our results can be applied to the strong cosmic censorship conjecture for the einstein maxwell real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data under christodoulou s c0 formulation the conjecture is proven to be false

a proof of the uniform boundedness of solutions to the wave equation on slowly rotating kerr backgrounds we consider kerr spacetimes with parameters a and m such that a m kerr newman spacetimes with parameters q m a m and more generally stationary axisymmetric black hole exterior spacetimes mathcal m g which are sufficiently close to a schwarzschild metric with parameter m 0 and whose killing fields span the null generator of the event horizon we show uniform boundedness on the exterior for solutions to the wave equation g 0 the most fundamental statement is at the level of energy we show that given a suitable foliation then there exists a constant c depending only on the parameter m and the choice of the foliation such that for all solutions a suitable energy flux through is bounded by c times the initial energy flux through 0 this energy flux is positive definite and does not degenerate at the horizon i e it agrees with the energy as measured by a local observer it is shown that a similar boundedness statement holds for all higher order energies again without degeneration at the horizon this leads in particular to the pointwise uniform boundedness of in terms of a higher order initial energy on 0 note that in view of the very general assumptions the separability properties of the wave equation or geodesic flow on the kerr background are not used in fact the physical mechanism for boundedness uncovered in this paper is independent of the dispersive properties of waves in the high frequency geometric optics regime

a proof of price s law on schwarzschild black hole manifolds for all angular momenta price s law states that linear perturbations of a schwarzschild black hole fall off as t 2 ell 3 for t to infty provided the initial data decay sufficiently fast at spatial infinity moreover if the perturbations are initially static i e their time derivative is zero then the decay is predicted to be t 2 ell 4 we give a proof of t 2 ell 2 decay for general data in the form of weighted l 1 to l infty bounds for solutions of the regge wheeler equation for initially static perturbations we obtain t 2 ell 3 the proof is based on an integral representation of the solution which follows from self adjoint spectral theory we apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates

microlocal analysis of asymptotically hyperbolic and kerr de sitter spaces with an appendix by semyon dyatlov in this paper we develop a general systematic microlocal framework for the fredholm analysis of non elliptic problems including high energy or semiclassical estimates which is stable under perturbations this framework described in sect 2 resides on a compact manifold without boundary hence in the standard setting of microlocal analysis many natural applications arise in the setting of non riemannian b metrics in the context of melrose s b structures these include asymptotically de sitter type metrics on a blow up of the natural compactification kerr de sitter type metrics as well as asymptotically minkowski metrics the simplest application is a new approach to analysis on riemannian or lorentzian or indeed possibly of other signature conformally compact spaces such as asymptotically hyperbolic or de sitter spaces including a new construction of the meromorphic extension of the resolvent of the laplacian in the riemannian case as well as high energy estimates for the spectral parameter in strips of the complex plane these results are also available in a follow up paper which is more expository in nature vasy in uhlmann g ed inverse problems and applications inside out ii 2012 the appendix written by dyatlov relates his analysis of resonances on exact kerr de sitter space which then was used to analyze the wave equation in that setting to the more general method described here

endpoint strichartz estimates and global solutions for the nonlinear dirac equation abstract we prove endpoint strichartz estimates for the klein gordon and wave equations in mixed norms on the polar coordinates in three spatial dimensions as an application global wellposedness of the nonlinear dirac equation is shown for small data in the energy class with some regularity assumption for the angular variable

comparison theorems for splittings of matrices this paper investigates the comparisons of asymptotic rates of convergence of two iteration matrices on the basis of nonnegative matrix theory comparisons between two nonnegative splittings and between two parallel multisplitting methods are derived when the coefficient matrix a is hermitian positive semi definite comparison theorems about two p regular splittings and two parallel multisplitting methods are proved

hidden symmetries and decay for the wave equation on the kerr spacetime energy and decay estimates for the wave equation on the exterior region of slowly rotating kerr spacetimes are proved the method used is a generalisation of the vector field method that allows the use of higher order symmetry operators in particular our method makes use of the second order carter operator which is a hidden symmetry in the sense that it does not correspond to a killing symmetry of the spacetime

convergence of waveform relaxation methods for hermitian positive definite linear systems abstract this paper studies the multi splitting algorithm and the two stage iterative method for waveform relaxation wr method solving the initial value problems for linear systems of ordinary differential equations odes in the form y t ay t f t convergence of both cases are analyzed when the coefficient matrix is hermitian positive definite and the splittings are p regular finally numerical experiment is presented

phase space analysis on some black hole manifolds abstract the schwarzschild and reissner nordstrom solutions to einstein s equations describe space times which contain spherically symmetric black holes we consider solutions to the linear wave equation in the exterior of a fixed black hole space time of this type we show that for solutions with initial data which decay at infinity and at the bifurcation sphere a weighted l 6 norm in space decays like t 1 3 this weight vanishes at the event horizon but not at infinity to obtain this control we require only an loss of angular derivatives

convergence of non stationary parallel multisplitting methods for hermitian positive definite matrices non stationary multisplitting algorithms for the solution of linear systems are studied convergence of these algorithms is analyzed when the coefficient matrix of the linear system is hermitian positive definite asynchronous versions of these algorithms are considered and their convergence investigated

dynamic iteration methods applied to linear dae systems abstract we continue the study of the convergence of dynamic iteration methods by applying them to linear dae systems we show that convergence rate can be studied by similar means as for ode s and that it is critical for convergence to preserve the structure of the dae system when it is split for the iteration

late time asymptotics for the wave equation on spherically symmetric stationary spacetimes abstract we derive precise late time asymptotics for solutions to the wave equation on spherically symmetric stationary and asymptotically flat spacetimes including as special cases the schwarzschild and reissner nordstrom families of black holes we also obtain late time asymptotics for the time derivatives of all orders and for the radiation field along null infinity we show that the leading order term in the asymptotic expansion is related to the existence of the conserved newman penrose quantities on null infinity as a corollary we obtain a characterization of all solutions which satisfy price s polynomial law 3 as a lower bound our analysis relies on physical space techniques and uses the vector field approach for almost sharp decay estimates introduced in our companion paper in the black hole case our estimates hold in the domain of outer communications up to and including the event horizon our work is motivated by the stability problem for black hole exteriors and strong cosmic censorship for black hole interiors

on a generalization of the richardson extrapolation process a convergence result for a generalized richardson extrapolation process is improved upon considerably and additional results of interest are proved an application of practical importance is also given finally some known results concerning the convergence of levin s t transformation are reconsidered in light of the results of the present work

an analysis of convergence for two stage waveform relaxation methods this paper consider a two stage or inner outer strategy for waveform relaxation wr iterations applied to initial value problems for linear systems of ordinary differential equations odes in the form y t qy t f t outer wr iterations are defined by yk 1 t dyk 1 t n1yk t f t where q d n1 and each iteration yk 1 t is computed using an inner iterative process based on an other splitting d m n2 each ode is then discretized by means of theta method for an m matrix q we prove that the method converges under the assumption that the whole splitting q m n1 n2 is an m splitting independently of the number of inner iterations moreover some comparison results are given in order to relate the ratio of convergence of the whole inner outer process both to the number of inner iterations actually done and to discretization parameters h and finally numerical experiments are presented

an algorithm for a generalization of the richardson extrapolation process in this paper we present a recursive method designated the w m algorithm for implementing a generalization of the richardson extrapolation process that has been ntroduced in 8 compared to the direct solution of the linear systems of equations defining the extrapolation procedure this method requires a small number of arithmetic operations and very little storage the technique is also applied to solve recursively the coefficient problem associated with the rational approximations obtained by applying the d transformation of 6 13 to power series in the course of development a new recursive algorithm for implementing a very general extrapolation procedure is introduced which is similar to that given in 2 4 for solving the same problem a fortran program for the w m algorithm is also appended

the n particle scattering problem asymptotic completeness for short range systems on demontre la completude asymptotique pour des systemes de mecanique quantique a courte portee constitues d un nombre arbitraire de particules

uniform decay of local energy and the semi linear wave equation on schwarzschild space we provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a schwarzschild background our estimate implies that such solutions have asymptotic behavior phi o left r 1 big t r big frac 1 2 right as long as the source term is bounded in the norm 1 frac 2m r 1 cdot 1 t r 1 l 1 big h 3 omega r 2dr d omega big in particular this gives scattering at small amplitudes for non linear scalar fields of the form square g varphi lambda varphi p varphi for all 2 p

the sign regularity of the auxiliary family g i x x a i ln x b i in convergence acceleration processes using the e algorithm abstract the e algorithm can be successfully applied to accelerate the convergence of the solution of some numerical differential equations provided the respective error expansion satisfies a certain asymptotic determinantal condition in this work it is proved that such condition is fulfilled for any error expansion in which formation enter the auxiliary family h n i ln h n i where h n designates the stepsize of the discretization method

two stage waveform relaxation method for the initial value problems of differential algebraic equations in this paper we consider a two stage strategy for waveform relaxation wr iterations applied to initial value problems for differential algebraic equations daes in the form a y t b y t f t outer iterations of tswr are defined by m a y k 1 t m 1 y k 1 t n 1 y k t n a y k t f t where a m a n a b m 1 n 1 and each iteration y k 1 t is computed using an inner iterative process based on another splitting m 1 m 2 n 2 meanwhile by the means of the theta method the discretized tswr of daes is realized furthermore when m a is an hermitian positive semi definite matrix with p regular splittings the convergence and the comparison theorems of tswr are analyzed finally the numerical experiments are presented

two stage iterations based on composite splittings for rectangular linear systems abstract in this paper we introduce a two stage method to solve rectangular linear systems that exhibits faster convergence than typical stationary iterative methods under suitable conditions we prove convergence of the new method the number of outer iterations can be reduced by using a few significant number of inner iterations for efficient computations further we perform a comparison analysis and establish that a higher number of inner iterations ensures a smaller spectral radius of the global iteration matrix we also discuss the uniqueness of a proper splitting and illustrate different comparison theorems for different subclasses of proper splittings

asymptotic upper bounds for the errors of richardson extrapolation with practical application in approximate computations the results produced by richardson extrapolation though in general very accurate are inexact numerical evaluation of this inexactness and implementation of the evaluation in practice are the objectives of this paper first considering linear changes of errors in the convergence plots asymptotic upper bounds are proposed for the errors then the achievement is extended to the results produced by richardson extrapolation and finally an error controlling procedure is proposed and successfully implemented in approximate computations originated in science and engineering copyright 2009 john wiley sons ltd

richardson extrapolation in boundary value problems for differential equations with nonregular right hand side abstract when the finite difference method is used to solve initial or boundary value problems with smooth data functions the accuracy of the numerical results may be considerably improved by acceleration techniques like richardson extrapolation however the success of such a technique is doubtful in cases were the right hand side or the coefficients of the equation are not sufficiently smooth because the validity of an asymptotic error expansion which is the theoretical prerequisite for the convergence analysis of the richardson extrapolation is not a priori obvious in this work we show that the richardson extrapolation may be successfully applied to the finite difference solutions of boundary value problems for ordinary second order linear differential equations with a nonregular right hand side we present some numerical results confirming our conclusions

the interior of charged black holes and the problem of uniqueness in general relativity we consider a spherically symmetric double characteristic initial value problem for the real einstein maxwell scalar field equations on the initial outgoing characteristic the data is assumed to satisfy the price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat cauchy surface we establish that the heuristic mass inflation scenario put forth by israel and poisson is mathematically correct in the context of this initial value problem in particular the maximal future development has a future boundary over which the space time is extendible as a c0 metric but along which the hawking mass blows up identically thus the space time is inextendible as a c1 metric in view of recent results of the author in collaboration with i rodnianski which rigorously establish the validity of price s law as an upper bound for the decay of scalar field hair the c0 extendibility result applies to the collapse of complete asymptotically flat spacelike initial data where the scalar field is compactly supported this shows that under christodoulou s c0 formulation the strong cosmic censorship conjecture is false for this system 2005 wiley periodicals inc

late time asymptotics for the wave equation on extremal reissner nordstrom backgrounds abstract we derive the precise late time asymptotics for solutions to the wave equation on extremal reissner nordstrom black holes and explicitly express the leading order coefficients in terms of the initial data our method is based on purely physical space techniques we derive novel weighted energy hierarchies and develop a singular time inversion theory which allow us to uncover the subtle contribution of both the near horizon and near infinity regions to the precise asymptotics we introduce a new horizon charge and provide applications pertaining to the interior dynamics of extremal black holes our work confirms and in some cases extends the numerical and heuristic analysis of lucietti murata reall tanahashi ori sela and blaksley burko