nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

unbalanced optimal transport dynamic and kantorovich formulations abstract this article presents a new class of distances between arbitrary nonnegative radon measures inspired by optimal transport these distances are defined by two equivalent alternative formulations i a dynamic formulation defining the distance as a geodesic distance over the space of measures ii a static kantorovich formulation where the distance is the minimum of an optimization problem over pairs of couplings describing the transfer transport creation and destruction of mass between two measures both formulations are convex optimization problems and the ability to switch from one to the other depending on the targeted application is a crucial property of our models of particular interest is the wasserstein fisher rao metric recently introduced independently by 7 15 defined initially through a dynamic formulation it belongs to this class of metrics and hence automatically benefits from a static kantorovich formulation

weighted estimates for a convolution appearing in the wave equation of hartree type abstract the paper concerns the cauchy and scattering problem of the wave equation of hartree type with small initial data with fast decay we prove weighted estimates for a convolution which appears in the equation

beyond the heisenberg time semiclassical treatment of spectral correlations in chaotic systems with spin 1 2 the two point correlation function of chaotic systems with spin 1 2 is evaluated using periodic orbits the spectral form factor for all times thus becomes accessible equivalence with the predictions of random matrix theory for the gaussian symplectic ensemble is demonstrated a duality between the underlying generating functions of the orthogonal and symplectic symmetry classes is semiclassically established

semiclassical form factor for chaotic systems with spin we study the properties of the two point spectral form factor for classically chaotic systems with spin in the semiclassical limit with a suitable semiclassical trace formula as our principal tool to this end we introduce a regularized form factor and discuss the limit in which the so called diagonal approximation can be recovered the incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics provided this skew product is mixing we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory

geometric properties of cones with applications on the hellinger kantorovich space and a new distance on the space of probability measures abstract we study general geometric properties of cone spaces and we apply them on the hellinger kantorovich space m x hk we exploit a two parameter scaling property of the hellinger kantorovich metric hk and we prove the existence of a distance shk on the space of probability measures that turns the hellinger kantorovich space m x hk into a cone space over the space of probabilities measures p x shk we provide a two parameter rescaling of geodesics in m x hk and for p x shk we obtain a full characterization of the geodesics we finally prove finer geometric properties including local angle condition and partial k semiconcavity of the squared distances that will be used in a future paper to prove existence of gradient flows on both spaces

weighted l and l1 estimates for solutions to the classical wave equation in three space dimensions on considere le probleme aux valeurs initiales u 0 u x 0 0 u t x 0 g x on donne des estimations l et l 1 pour les solutions dans le cas general a symetrie non spherique

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

the bakerian lecture 1987 quantum chaology bounded or driven classical systems often exhibit chaos exponential instability that persists but their quantum counterparts do not nevertheless there are new regimes of quantum behaviour that emerge in the semiclassical limit and depend on whether the classical orbits are regular or chaotic and this motivates the following definition definition quantum chaology is the study of semiclassical but nonclassical behaviour characteristic of systems whose classical motion exhibits chaos this is illustrated by the statistics of energy levels on scales comparable with the mean level spacing of order h n for n freedoms these fall into universality classes for classically chaotic systems the statistics are those of random matrices real symmetric or complex hermitian depending on the presence or absence of time reversal symmetry for classically regular ones the statistics are poisson on larger scales of order h i e classically small but semiclassically large universality breaks down these phenomena are being explained by representing spectra in terms of classical closed orbits universal spectral behaviour has its origin in very long orbits non universal behaviour depends only on short ones

gradient flows and evolution variational inequalities in metric spaces i structural properties abstract this is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space x d that can be characterized by evolution variational inequalities evi we present new results concerning the structural properties of solutions to the evi formulation such as contraction regularity asymptotic expansion precise energy identity stability asymptotic behavior and their link with the geodesic convexity of the driving functional under the crucial assumption of the existence of an evi gradient flow we will also prove two main results the equivalence with the de giorgi variational characterization of curves of maximal slope the convergence of the minimizing movement jko scheme to the evi gradient flow with an explicit and uniform error estimate of order 1 2 with respect to the step size independent of any geometric hypothesis as upper or lower curvature bounds on d in order to avoid any compactness assumption we will also introduce a suitable relaxation of the minimizing movement algorithm obtained by the ekeland variational principle and we will prove its uniform convergence as well

convex sobolev inequalities related to unbalanced optimal transport abstract we study the behavior of various lyapunov functionals relative entropies along the solutions of a family of nonlinear drift diffusion reaction equations coming from statistical mechanics and population dynamics these equations can be viewed as gradient flows over the space of radon measures equipped with the hellinger kantorovich distance the driving functionals of the gradient flows are not assumed to be geodesically convex or semi convex we prove new isoperimetric type functional inequalities allowing us to control the relative entropies by their productions which yields the exponential decay of the relative entropies

resummation and the semiclassical theory of spectral statistics we address the question as to why in the semiclassical limit classically chaotic systems generically exhibit universal quantum spectral statistics coincident with those of random matrix theory to do so we use a semiclassical resummation formalism that explicitly preserves the unitarity of the quantum time evolution by incorporating duality relations between short and long classical orbits this allows us to obtain both the non oscillatory and the oscillatory contributions to spectral correlation functions within a unified framework thus overcoming a significant problem in previous approaches in addition our results extend beyond the universal regime to describe the system specific approach to the semiclassical limit

scattering theory for semilinear wave equations with small data in two space dimensions we study scattering theory for the semilinear wave equation utt au jujp lu in two space dimensions we show that if p p0 3 x ii 2 the scattering operator exists for smooth and small data the lower bound p0 of p is considered to be optimal see glassey 6 7 schaeffer 18 our result is an extension of the results by strauss 19 klainerman 10 and mochizuki and motai 14 15 the construction of the scattering operator for small data does not follow directly from the proofs in 7 13 20 and 22 concerning the global existence of solutions for the cauchy problem of the above equation with small initial data given at t 0 in two space dimensions because we have to consider the integral equation with unbounded integral region associated to the above equation u x t u x t i ii s dy ds 27r ij jix yl t s ts 2 ix y12 for t e r where u x t is a solution of utt au 0 which u x t approaches asymptotically as t oo the proof of the basic estimate for the above integral equation is more difficult and complicated than that for the cauchy problem of utt au iu pu in two space dimensions

scattering for semilinear wave equations with small data in three space dimensions on considere la theorie de la diffusion pour des equations d onde semilineaires du type u tt u f u a 3 dimensions d espace ou f u u p 1 u ou f u u p r p 1

gradient flows on wasserstein spaces over compact alexandrov spaces we establish the existence of euclidean tangent cones on wasserstein spaces over compact alexandrov spaces of curvature bounded below by using this riemannian structure we formulate and construct gradient flows of functions on such spaces if the underlying space is a riemannian manifold of nonnegative sectional curvature then our gradient flow of the free energy produces a solution of the linear fokker planck equation

periodic orbit resummation and the quantization of chaos we study the spectral determinant e which has by construction zeros at the quantum energy levels of a given system if the classical motion of the system in question is chaotic then e has a semiclassical representation as a sum over combinations of periodic orbits there are however a number of fundamental problems associated with its convergence properties imposing upon the sum the condition that like e itself it is real for real e we obtain formal resummation equations relating the contributions from asymptotically long orbits to those of the short orbits these then lead to a formal derivation of the previously conjectured riemann siegel lookalike formula which involves only a finite orbit sum and thus represents in principle a semiclassical rule for quantizing chaos

optimal entropy transport problems and a new hellinger kantorovich distance between positive measures we develop a full theory for the new class of optimal entropy transport problems between nonnegative and finite radon measures in general topological spaces these problems arise quite naturally by relaxing the marginal constraints typical of optimal transport problems given a pair of finite measures with possibly different total mass one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals which quantify in some way the deviation of the marginals of the transport plan from the assigned measures as a powerful application of this theory we study the particular case of logarithmic entropy transport problems and introduce the new hellinger kantorovich distance between measures in metric spaces the striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance which lies somehow between the well known hellinger kakutani and kantorovich wasserstein distances

two point correlations of the gaussian symplectic ensemble from periodic orbits we determine the asymptotics of the two point correlation function for quantum systems with half integer spin which show chaotic behaviour in the classical limit using a method introduced by bogomolny and keating for time reversal invariant systems we obtain the leading terms of the two point correlation function of the gaussian symplectic ensemble special attention has to be paid to the role of kramers degeneracy

the spin contribution to the form factor of quantum graphs following the quantization of a graph with the dirac operator spin 1 2 we explain how additional weights in the spectral form factor k due to spin propagation around orbits produce higher order terms in the small asymptotics in agreement with symplectic random matrix ensembles we determine conditions on the group of spin rotations sufficient to generate cse statistics

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

the asymptotic behavior of yang mills fields in the large we consider yang mills fields in minkowski space time and prove that all spherically symmetric solutions in the canonical gauge decay in time provided the initial data has finite conformal energy

high orders of the weyl expansion for quantum billiards resurgence of periodic orbits and the stokes phenomenon a formalism is developed for calculating high coefficients c r of the weyl high energy expansion for the trace of the resolvent of the laplace operator in a domain b with smooth boundary b the c r are used to test the following conjectures a the sequence of c r diverges factorially controlled by the shortest accessible real or complex periodic geodesic b if this is a 2 bounce orbit it corresponds to the saddle of the chord length function whose contour is first crossed when climbing from the diagonal of the mobius strip which is the space of chords of b c this orbit gives an exponential contribution to the remainder when the weyl series truncated at its least term is subtracted from the resolvent the exponential switches on smoothly according to an error function where it is smallest that is across the negative energy axis stokes line these conjectures are motivated by recent results in asymptotics they survive tests for the circle billiard and for a family of curves with 2 and 3 bulges where the dominant orbit is not always the shortest and is sometimes complex for some systems which are not smooth billiards e g a particle on a ring or in a billiard where b is a polygon the weyl series terminates and so no geodesics are accessible for a particle on a compact surface of constant negative curvature only the complex geodesics are accessible from the weyl series

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

semiclassical approach to parametric spectral correlation with spin 1 2 the spectral correlation of a chaotic system with spin 1 2 is universally described by the gse gaussian symplectic ensemble of random matrices in the semiclassical limit in semiclassical theory the spectral form factor is expressed in terms of the periodic orbits and the spin state is simulated by the uniform distribution on a sphere in this paper instead of the uniform distribution we introduce brownian motion on a sphere to yield the parametric motion of the energy levels as a result the small time expansion of the form factor is obtained and found to be in agreement with the prediction of parametric random matrices in the transition within the gse universality class moreover by starting the brownian motion from a point distribution on the sphere we gradually increase the effect of the spin and calculate the form factor describing the transition from the gaussian orthogonal ensemble class to the gse class

monge kantorovitch measure transportation and monge ampere equation on wiener space let w h be an abstract wiener space assume two i i 1 2 probabilities on w w we give some conditions for the wasserstein distance between 1 and 2 with respect to the cameron martin space to be finite where the infimum is taken on the set of probability measures on w w whose first and second marginals are 1 and 2 in this case we prove the existence of a unique cyclically monotone map t iw with w h such that t maps 1 to 2 moreover if 2 then t is stochastically invertible i e there exists s w w such that s t iw 1 a s and t s iw 2 a s if in addition 1 then there exists a 1 convex function in the gaussian sobolev space such that these results imply that the quasi invariant transformations of the wiener space with finite wasserstein distance from can be written as the composition of a transport map t and a rotation i e a measure preserving map we give also 1 convex sub solutions and ito type solutions of the monge ampere equation on w

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

critical blowup for systems of semilinear wave equations in low space dimensions abstract we consider the cauchy problem for systems of semilinear wave equations in two and three space dimensions with small initial data del santo et al geometric optics and related topics f colombini and n lerner eds progress in nonlinear differential equations and their applications vol 32 pp 117 140 birkhauser boston 1997 have studied the existence and nonexistence of global classical solutions of the cauchy problem except for the critical case in this paper we study the critical case and we show the nonexistence of global classical solutions and also give the upper bounds of the life span

a rule for quantizing chaos the authors find a real function delta e whose zeros approximate the quantum energy levels of a system with chaotic classical trajectories delta e is a finite sum over combinations of classical periodic orbits it is obtained from gutzwiller s infinite and divergent sum 1982 representing the spectral density in terms of periodic orbits by means of a resummation conjectured by analogy with a derivation of the riemann siegel formula for the riemann zeros they assess the practicality of the quantization condition

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

the semiclassical origin of the logarithmic singularity in the symplectic form factor sieber and richter achieved a breakthrough towards a proof of the universality of spectral fluctuations of chaotic quantum systems conjectured by bohigas giannoni and schmidt by calculating semiclassically the first term beyond the diagonal approximation of the orthogonal form factor in this letter the semiclassical origin of the logarithmic singularity of the symplectic form factor is deduced perturbatively by combining this result with the contribution that arises due to the spin this approach stands in contrast to the duality approach introduced by bogomolny and keating which is essentially non perturbative and where the structure around the heisenberg time is related to the structure for very small time which can be deduced using the diagonal approximation

l p estimates for the linear wave equation and global existence for semilinear wave equations in exterior domains we consider the initial boundary value problem for the semilinear wave equation u tt delta u a x u t f u mbox in omega times 0 infty u x 0 u 0 x u t x 0 u 1 x mbox and u partial omega 0 where omega is an exterior domain in r n a x u t is a dissipative term which is effective only near the critical part of the boundary we first give some l p estimates for the linear equation by combining the results of the local energy decay and l p estimates for the cauchy problem in the whole space next on the basis of these estimates we prove global existence of small amplitude solutions for semilinear equations when omega is odd dimensional domain when n 3 mbox and f u alpha u our result is applied if alpha 2 sqrt 3 1 we note that no geometrical condition on the boundary partial omega is imposed