polynomial bound on the number of scattering poles on montre que dans la diffusion par un potentiel infiniment differentiable a support compact les poles de l operateur de diffusion sont a distribution polynomiale

totally geodesic foliations on 4 manifolds we give a rather detailed description of the behavior of 2 dimensional totally geodesic foliations on compact riemannian 4 manifolds in particular we obtain a complete characterization in the simply connected case

locally convex spaces containing no copy of c0 abstract we show that locally cinvex spaces which contain no copy of c 0 have a series of very nice properties

the geometry of symplectic pairs we study the geometry of manifolds carrying symplectic pairs consisting of two closed 2 forms of constant ranks whose kernel foliations are complementary using a variation of the construction of boothby and wang we build contact symplectic and contact pairs from symplectic pairs

spectral behavior of contractive noise we study the behavior of the spectra corresponding to quantum systems subjected to a contractive noise i e the environment reduces the accessible phase space of the system but the total probability is conserved we find that the number of long lived resonances grows as a power law in h but surprisingly there is no relationship between the exponent of this power law and the fractal dimension of the corresponding classical attractor this is in disagreement with the predictions of the fractal weyl law which has been established for open systems where the probability is lost under the effect of a projective opening

banach limits and uniform almost summability abstract some classical hahn schur theorem like results on the uniform convergence of unconditionally convergent series can be generalized to weakly unconditionally cauchy series in this paper we obtain this type of generalization via a summability method based upon the concept of almost convergence we also obtain a generalization of the main result in aizpuru et al 2003 3 using pointwise convergence of sums indexed in natural boolean algebras with the vitali hahn saks property in order to achieve that we first study the notion of almost convergence through its original definition which involves banach limits giving a description of the extremal structure of the set of all norm 1 hahn banach extensions of the limit function on c to l we also show the existence of norm 1 hahn banach extensions of the limit function on c to l that are not extensions of the almost limit function and hence are not banach limits

weyl asymptotics from closed to open systems we present microwave experiments on the symmetry reduced five disk billiard studying the transition from a closed to an open system the measured microwave reflection signal is analyzed by means of the harmonic inversion and the counting function of the resulting resonances is studied for the closed system this counting function shows the weyl asymptotic with a leading exponent equal to 2 by opening the system successively this exponent decreases smoothly to a noninteger value for the open systems the extraction of resonances by the harmonic inversion becomes more challenging and the arising difficulties are discussed the results can be interpreted as a first experimental indication for the fractal weyl conjecture for resonances

boolean algebras and uniform convergence of series abstract several classical results on uniform convergence of unconditionally cauchy series are generalized to weakly unconditionally cauchy series this uniform convergence is characterized through subalgebras and subfamilies of p n a generalization of the orlicz pettis theorem is also proved by mean of subalgebras of p n

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

complex scaling and the distribution of scattering poles the purpose of this paper is to establish sharp polynomial bounds on the number of scattering poles for a general class of compactly supported self adjoint perturbations of the laplacian in r n odd we also consider more general types of bounds that give sharper estimates in certain situations the general conclusion can be stated as follows the order of growth of the poles is the same as the order of growth of eigenvalues for corresponding compact problems from the few known cases however the exact asymptotics are expected to be different the scattering poles for compactly supported perturbations were rigorously defined by lax and phillips 13 in a more general setting they correspond to resonances the study of which has a long tradition in mathematical physics in the lax phillips theory they appear as the poles of the meromorphic continuation of the scattering matrix and coincide with the poles of the meromorphic continuation of the resolvent of the perturbed operator because of the latter characterization they can be considered as the analogue of the discrete spectral data for problems on noncompact domains the problem of estimating the counting function

geometric bounds on the density of resonances for semiclassical problems in this paper we shall give upper bounds on the number of resonances in certain regions in the complex plane close to the real axis for semiclassical operators like h 2 v x when h is small

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

on the location of resonances for schrodinger operators in the semiclassical limit i resonances free domains abstract we show that under suitable non trapping conditions on the potential v at energy e the schrodinger operator h h 2 v exhibits no resonance in a neighbourhood of e for small h we briefly discuss some critical values of e like thresholds or barrier tops

pseudospectra of semiclassical pseudo differential operators the pseudo spectra or spectral instability of non selfadjoint operators is a topic of current interest in applied mathematics for example in computational fluid dynamics it affects the study of the stability of laminar flows in fact even for the most basic flows the computations entirely fails to predict what is observed in the experiments the explanation is that for non normal operators the resolvent could be very large far away from the spectrum which makes computation of the eigenvalues impossible the occurence of false eigenvalues is due to the existence of quasi modes i e approximate local solutions to the eigenvalue problem the quasi modes appear since the nirenberg treves condition psi is not satisfied for topological reasons less

quantum decay rates in chaotic scattering we study quantum scattering on manifolds equivalent to the euclidean space near infinity in the semiclassical regime we assume that the corresponding classical flow admits a non trivial trapped set and that the dynamics on this set is of axiom a type uniformly hyperbolic we are interested in the distribution of quantum resonances near the real axis in two dimensions we prove that if the trapped set is sufficiently thin then there exists a gap between the resonances and the real axis that is quantum decay rates are bounded from below in higher dimension the condition for this gap is given in terms of a certain topological pressure associated with the classical flow under the same assumption we also prove a resolvent estimate with a logarithmic loss compared to non trapping situations

decay of correlations for normally hyperbolic trapping we prove that for evolution problems with normally hyperbolic trapping in phase space correlations decay exponentially in time normally hyperbolic trapping means that the trapped set is smooth and symplectic and that the flow is hyperbolic in directions transversal to it flows with this structure include contact anosov flows classical flows in molecular dynamics and null geodesic flows for black holes metrics the decay of correlations is a consequence of the existence of resonance free strips for green s functions cut off resolvents and polynomial bounds on the growth of those functions in the semiclassical parameter

scattering asymptotics for riemann surfaces in this article we prove the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends we also give weyl asymptotics for the relative scattering phase of such a surface the proofs are based on trace formulae analogous to those of the euclidean odd dimensional scattering the main technical ingredient is a new proof of the poisson formula theorem 5 7 which is applicable in the euclidean case as well our lower bound seems to be the first example of an optimal polynomial lower bound for the number of resonances holding for a general class of higher dimensional elliptic operators with no symmetries the previous general lower bounds or asymptotics were either nonoptimal 25 58 9 one dimensional or radial 65 67 and 54 41 1 or they required some degeneracy of the

local smoothing for scattering manifolds with hyperbolic trapped sets we prove a resolvent estimate for the laplace beltrami operator on a scattering manifold with a hyperbolic trapped set and as a corollary deduce local smoothing we use a result of nonnenmacher zworski to provide an estimate near the trapped region a result of burq and cardoso vodev to provide an estimate near infinity and the microlocal calculus on scattering manifolds to combine the two

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

distribution of poles for scattering on the real line abstract the density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential in the case of a potential with finite singularities at the endpoints of the support asymptotic formulae for the poles are given while in the c 0 case an example of a potential with infinitely many scattering poles on i r is constructed the scattering amplitude of a compactly supported potential is also characterized

fractal weyl law for open quantum chaotic maps we study a semiclassical quantization of poincar e maps arising in scattering problems with fractal hyperbolic trapped sets the main application is the proof of a fractal weyl upper bound for the number of resonances scattering poles in small domains near the real axis this result includes the case of several convex hard obstacles satisfying a no eclipse condition

resonances en limite semiclassique et exposants de lyapunov we determine the width of resonance free domains in the complex plane for the semiclassical schrodinger operator h2 v x whenh 0 in terms of lyapunov exponents for the associated classical flow

the fbi transform on compact manifolds we present a geometric theory of the fourier bros iagolnitzer transform on a compact c manifold m the fbi transform is a generalization of the classical notion of the wave packet transform we discuss the mapping properties of the fbi transform and its relationship to the calculus of pseudodifferential operators on m we also describe the microlocal properties of its range in terms of the scattering calculus of pseudodifferential operators on the noncompact manifold t m

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

quasimodes and resonances sharp lower bounds we prove that asymptotically any cluster of quasimodes close to each other approximates at least the same number of resonances counting multiplicities as a consequence we get that the counting function of the number of resonances close to the real axis is bounded from below essentially by the counting function of the quasimodes

distribution of resonances for open quantum maps we analyze a simple model of quantum chaotic scattering system namely the quantized open baker s map this model provides a numerical confirmation of the fractal weyl law for the semiclassical density of quantum resonances the fractal exponent is related to the dimension of the classical repeller we also consider a variant of this model for which the full resonance spectrum can be rigorously computed and satisfies the fractal weyl law for that model we also compute the shot noise of the conductance through the system and obtain a value close to the prediction of random matrix theory

schrodinger operators with complex valued potentials and no resonances in dimension d 3 we give examples of nontrivial compactly supported complex valued potentials such that the associated schrodinger operators have neither resonances nor eigenvalues if d 2 we show that there are potentials with no resonances or eigenvalues away from the origin these schrodinger operators are isophasal and have the same scattering phase as the laplacian on rd in odd dimensions d 3 we study the fundamental solution of the wave equation perturbed by such a potential if the space variables are held fixed it is superexponentially decaying in time

unconditionally convergent series of operators on banach spaces abstract we show that any series n knof operators inl x y that is unconditionally convergent in the weak operator topology and satisfies the condition that n f knis a compact operator for every index setf n is unconditionally convergent in the uniform operator topology if and only ifx the dual space of the banach spacex contains no copy ofc0

breit wigner approximation and the distribution of resonances for operators with a discrete spectrum j 2 the counting function of j s n trivially satisfies n n j j in scattering situations the natural analogue of the discrete spectrum is given by resonances j and of n by the scattering phase s the relation between the two is now non trivial and we prove that where is the harmonic measure of the upper of half plane and can be taken dependent on this provides a precise high energy version of the breit wigner approximation and relates the properties of s to the distribution of resonances close to the real axis

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

probabilistic weyl laws for quantized tori for the toeplitz quantization of complex valued functions on a 2n dimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural weyl law 1 3 in numerical experiments the same weyl law also holds for false eigenvalues created by pseudospectral effects

scattering phase asymptotics with fractal remainders for a riemannian manifold m g which is isometric to the euclidean space outside of a compact set and whose trapped set has liouville measure zero we prove weyl type asymptotics for the scattering phase with remainder depending on the classical escape rate and the maximal expansion rate for axiom a geodesic flows this gives a polynomial improvement over the known remainders we also show that the remainder can be bounded above by the number of resonances in some neighbourhoods of the real axis and provide similar asymptotics for hyperbolic quotients using the selberg zeta function

the geometry of recursion operators we study the fields of endomorphisms intertwining pairs of symplectic structures using these endomorphisms we prove an analogue of moser s theorem for simultaneous isotopies of two families of symplectic forms we also consider the geometric structures defined by pairs and triples of symplectic forms for which the squares of the intertwining endomorphisms are plus or minus the identity for pairs of forms we recover the notions of symplectic pairs and of holomorphic symplectic structures for triples we recover the notion of a hypersymplectic structure and we also find three new structures that have not been considered before one of these is the symplectic formulation of hyper kahler geometry which turns out to be a strict generalization of the usual definition in terms of differential or kahler geometry

semiclassical resonances generated by a closed trajectory of hyperbolic type we determine all the resonances in certain rectangular regions of the complex plane of the schrodinger operator h2 v whenh 0 under the assumption that the set of trapped points of energy 0 for the classical flow form a closed trajectory and that the corresponding poincare map is hyperbolic

resonance expansions of propagators in the presence of potential barriers abstract we show that the schrodinger propagator can be expanded in terms of resonances at energy levels at which a barrier separates the interaction region from infinity the expansions hold for all times with errors small in the semi classical parameter as a byproduct we obtain a result on the approximation of clusters of resonant states by clusters of eigenfunctions of a self adjoint reference operator

the selberg zeta function for convex co compact schottky groups we give a new upper bound on the selberg zeta function for a convex co compact schottky group acting on the hyperbolic space n 1 in strips parallel to the imaginary axis the zeta function is bounded by exp c s where is the dimension of the limit set of the group this bound is more precise than the optimal global bound exp c s n 1 and it gives new bounds on the number of resonances scattering poles of n 1 the proof of this result is based on the application of holomorphic l2 techniques to the study of the determinants of the ruelle transfer operators and on the quasi self similarity of limit sets we also study this problem numerically and provide evidence that the bound may be optimal our motivation comes from molecular dynamics and we consider n 1 as the simplest model of quantum chaotic scattering

hyperbolic manifolds are geodesically rigid we show that if all geodesics of two non proportional metrics on a closed manifold coincide as unparameterized curves then the manifold has a finite fundamental group or admits a local product structure this implies that if the manifold admits a metric of negative sectional curvature then two metrics on the manifold have the same geodesics if and only if they are proportional

local smoothing for the schrodinger equation with a prescribed loss we consider a family of surfaces of revolution each with a single periodic geodesic which is degenerately unstable we prove a local smoothing estimate for solutions to the linear schrodinger equation with a loss that depends on the degeneracy and we construct explicit examples to show our estimate is saturated on a weak semiclassical time scale as a byproduct of our proof we obtain a cutoff resolvent estimate with a sharp polynomial loss

density and location of resonances for convex co compact hyperbolic surfaces let x vargamma backslash mathbb h 2 be a convex co compact hyperbolic surface and let be the hausdorff dimension of the limit set let x be the hyperbolic laplacian we show that the density of resonances of the laplacian x in rectangles bigl sigma leq mathrm re s leq delta big vert mathrm im s big vert leq t bigr is less than o t1 in the limit t where frac delta 2 this improves the previous fractal weyl upper bound of zworski invent math 136 2 353 409 1999 and goes in the direction of a conjecture stated in jakobson and naud geom funct anal 22 2 352 368 2012

traceformula for resonances in small domains abstract traceformulas are useful in the study of resonances in this article we work in a semi classical black box framework and obtain a traceformula of the same type as that of the second author now for the resonances in h dependent domains of size h and under somewhat more restrictive assumptions as an application we extend to a long range situation a result by v petkov and m zworski on breit wigner peaks in the scattering phase