recovering asymptotics of metrics from fixed energy scattering data from fixed energy scattering data is studied it is shown that if two such metrics g1 g2 have scattering data at some fixed energy which are equal up to smoothing then there exists a diffeomorphism fixing infinity such that g1 g2 is rapidly decreasing given the scattering matrix at two energies it is shown that the asymptotics of a metric and a short range potential can be determined simultaneously these results also hold for a wide class of scattering manifolds

resolvent at low energy iii the spectral measure the second and third authors were supported by australian research council discovery grants dp0771826 and dp1095448 and the second author by a future fellowship the first author was partially supported by anr grant anr 09 jcjc 0099 01 and by the pics cnrs progress in geometric analysis and applications and thanks the math department of anu for its hospitality

the schrodinger propagator for scattering metrics let g be a scattering metric on a compact manifold x with boundary i e a smooth metric giving the interior x the structure of a complete riemannian manifold with asymptotically conic ends an example is any compactly supported perturbation of the standard metric on rn consider the operator h 1 2 v where is the positive laplacian with respect to g and v is a smooth real valued function on x vanishing to second order at iox assuming that g is nontrapping we construct a global parametrix u z w t for the kernel of the schri odinger propagator u t e ith where z w io x and t 0 the parametrix is such that the difference between u and u is smooth and rapidly decreasing both as t iae 0 and as z iae iox uniformly for w on compact subsets of x let r x 1 where x is a boundary defining function for x be an asymptotic radial variable and let w t be the kernel e ir2 2tu t using the parametrix we show that w t belongs to a class of i legendre distributionsi on x i x i r 0 previously considered by hassell vasy when the metric is trapping then the parametrix construction goes through microlocally in the nontrapping part of the phase space we apply this result to determine the singularities of u t f for any tempered distribution f and for any fixed t 0 in terms of the oscillation of f near iox if the metric is nontrapping then we precisely determine the wavefront set of u t f and hence also precisely determine its singular support more generally we are able to determine the wavefront set of u t f for t 0 resp t 0 on the non backward trapped resp non forward trapped subset of the phase space this generalizes results of craig kappeler strauss and wunsch

the semiclassical resolvent and the propagator for non trapping scattering metrics abstract consider a compact manifold with boundary m with a scattering metric g or equivalently an asymptotically conic manifold m g euclidean r n with a compactly supported metric perturbation is an example of such a space let be the positive laplacian on m g and v a smooth potential on m which decays to second order at infinity in this paper we construct the kernel of the operator h 2 v 0 i 0 2 1 at a non trapping energy 0 0 uniformly for h 0 h 0 h 0 0 small within a class of legendre distributions on manifolds with codimension three corners using this we construct the kernel of the propagator e i t 2 v t 0 t 0 as a quadratic legendre distribution we also determine the global semiclassical structure of the spectral projector poisson operator and scattering matrix

sharp strichartz estimates on nontrapping asymptotically conic manifolds we obtain the strichartz inequalities u l q t l r x 0 1 m c u 0 l 2 m for any smooth n dimensional riemannian manifold m which is asymptotically conic at infinity with either short range or long range metric perturbation and nontrapping where u is a solution to the schrodinger equation iu t 1 2 m u 0 and 2 q r are admissible strichartz exponents 2 q n r n 2 this corresponds with the estimates available for euclidean space except for the endpoint q r 2 2 n n 2 when n 2 these estimates imply existence theorems for semi linear schrodinger equations on m by adapting arguments from cazenave and weissler and kato this result improves on our previous result which was an l 4 t x strichartz estimate in three dimensions it is closely related to results of staffilani tataru burq robbiano zuily and tataru who consider the case of asymptotically flat manifolds

scattering metrics and geodesic flow at infinity any compact manifold with boundary admits a riemann metric on its interior taking the form x 4 dx 2 x 2 h near the boundary where x is a boundary defining function and h is a smooth symmetric 2 cotensor restricting to be positive definite and hence a metric h on the boundary the scattering theory associated to the laplacian for such a scattering metric was discussed by the first author and here it is shown as conjectured that the scattering matrix is a fourier integral operator which quantizes the geodesic flow on the boundary for the metric h at time to prove this the poisson operator of the associated generalized boundary problem is constructed as a fourier integral operator associated to a singular legendre manifold

distribution of resonances for asymptotically euclidean manifolds in this paper we discuss meromorphic continuation of the resolvent and bounds on the number of resonances for scattering manifolds a class of manifolds generalizing euclidian n space subject to the basic assumption of analyticity near infinity we show that resolvent of the laplacian has a meromorphic continuation to a conic neighborhood of the continuous spectrum this involves a geometric interpretation of the complex scaling method in terms of deformations in the grauert tube of the manifold we then show that the number of resonances poles of the meromorphic continuation of the resolvent in a conic neighborhood of mathbb r of absolute value less than r 2 is mathcal o r n under the stronger assumption of global analyticity and hyperbolicity of the geodesic flow we prove a finer weyl type upper bound for the counting function for resonances in small neighborhoods of the real axis this estimate has an exponent which involves the dimension of the trapped set of the geodesic flow

on enclosing simple roots of nonlinear equations in this paper we present two efficient algorithms for enclosing a simple root of the nonlinear equation f x 0 in the interval a b they improve recent methods of alefeld and potra by achieving higher efficiency indices and avoiding the solution of a quadratic equation per iteration the efficiency indices of our methods are 1 5537 and 1 618 respectively we show that our second method is an optimal algorithm in some sense our numerical experiments show that the two methods of the present paper compare well with the above methods of alefeld and potra as well as efficient solvers of dekker brent and le the second method in this paper has the best behavior of all especially when the termination tolerance is small

smoothness and non smoothness of the fundamental solution of time dependent schrodinger equations the fundamental solutione t s x y of time dependent schrodinger equationsi u t 1 2 u v t x u is studied it is shown that e t s x y is smooth and bounded fort s if the potential is sub quadratic in the sense thatv t x o x 2 at infinity in one dimension ifv t x v x is time independent and super quadratic in the sense thatv x c 1 x 2 e at infinity c 0 and e 0 thene t s x y is nowherec1

parametrices and dispersive estimates for schrodinger operators with variable coefficients in this article we consider variable coefficient time dependent schr odinger evolutions in bbb r n using phase space methods we construct outgoing parametrices and prove strichartz type estimates globally in time this is done in the context of c 2 metrics which satisfy a weak asymptotic flatness condition at infinity

some substantial modifications and improvements for derivative free iterative methods and derivative free transformation for multiple zeros in this paper we present some substantial modifications and improvements for some derivative free iterative formulae we propose some new methods which are convergent iterative formulae of order 1 2 2 414 with only two evaluations of f per step thus the new methods enjoy the efficiency index 1 2 1 554 especially a new method of self accelerating regula falsi type with global convergence for finding a simple root p of a nonlinear equation f x 0 in the interval a b is offered in this paper the new method of self accelerating regula falsi type is shown to be convergent with order 1 2 2 414 for both the sequences of diameters b n a n and the iterative points x n the new method has been tested on a series of published examples the numerical results demonstrate that the new method is more effective furthermore we introduce a derivative free transformation for multiple zeros and consider its error analysis and then demonstrate its superiority by comparing it with the traditional ones

on open scattering channels for manifolds with ends abstract in the framework of time dependent geometric scattering theory we study the existence and completeness of the wave operators for perturbations of the riemannian metric for the laplacian on a complete manifold of dimension n the smallness condition for the perturbation is expressed intrinsically and coordinate free in purely geometric terms using the harmonic radius therefore the size of the perturbation can be controlled in terms of local bounds on the injectivity radius and the ricci curvature as an application of these ideas we obtain a stability result for the scattering matrix with respect to perturbations of the riemannian metric this stability result implies that a scattering channel which interacts with other channels preserves this property under small perturbations

strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifolds abstract we prove the global in time strichartz estimates for wave equations on the nontrapping asymptotically conic manifolds we obtain estimates for the full set of wave admissible indices including the endpoint the key points are the properties of the microlocalized spectral measure of laplacian on this setting showed in 18 and a littlewood paley squarefunction estimate as applications we prove the global existence and scattering for a family of nonlinear wave equations on this setting

reconstruction of singularities for solutions of schrodinger s equation we determine the behavior in time of singularities of solutions to some schrodinger equations onr n we assume the hamiltonians are of the formh 0 v where h 0 1 2 delta 1 2 sum limits k 1 n omega k 2 x k 2 and wherev is bounded and smooth with decaying derivatives when all k 0 the kernelk t x y of exp ith is smooth inx for every fixed t y when all 1 are equal but non zero the initial singularity reconstructs at times t frac m pi omega 1 and positionsx 1 m y just as ifv 0 k is otherwise regular in the general case the singular support is shown to be contained in the union of the hyperplanes x x js 1 l js y js when j t l j forj j 1 j r

approximation and equidistribution of phase shifts spherical symmetry consider a semiclassical hamiltonian h v h h 2 delta v e where h 0 is a semiclassical parameter is the positive laplacian on mathbb r d v is a smooth compactly supported central potential function and e 0 is an energy level in this setting the scattering matrix sh e is a unitary operator on l 2 mathbb s d 1 hence with spectrum lying on the unit circle moreover the spectrum is discrete except at 1

semiclassical estimates in asymptotically euclidean scattering we consider long range semiclassical perturbations of the laplacian on asymptotically euclidean manifolds we obtain precise resolvent estimates under non trapping assumptions the novelty lies in a systematic use of geometric microlocal methods

parametrices for a class of schrodinger equations the article describes the construction of a parametrix global in space in the initial value problem for the schrodinger equation i u t u q x u v x u with q a real quadratic form and v a real potential in the hypothesis on v is that v x const x and v x const x 1 as x by conjugating the equation with the unitary group exp ith h q x the problem is reduced to a hyperbolic pseudodifferential operator to which a global version of the geometric optics method applies the symbolic calculus used by the construction is based on weights that reflect the role of the propagation set

a fourth order derivative free algorithm for nonlinear equations a two step derivative free iterative algorithm is presented for solving nonlinear equations error analysis shows that the algorithm is fourth order with efficiency index equal to 1 5874 a lot of numerical results show that the algorithm is effective and is preferable to some existing derivative free methods in terms of computation cost

micro local resolvent estimates for 2 body schrodinger operators on considere l operateur de schrodinger h v x dans l 2 r n en relation avec des localisations dans l espace de phase r x n r n

quadratically convergent multiple roots finding method without derivatives abstract in this paper an iteration method without derivatives for multiple roots is proposed this method proved to be quadratically convergent its efficiency and accuracy are illustrated by numerical experiments