on the contribution of degenerate periodic trajectories to the wave trace this paper is concerned with the wave trace z t of a selfadjoint elliptic pseudodifferential operator on a compact manifold it is devoted to the contribution of degenerate periodic trajectories of the associated hamiltonian flow to the singularities of z t given a periodic trajectory and a point on it we obtain a trace formula which relates microlocally the trace of the unitary group of a in a neighborhood of to an oscillatory integral the phase and the amplitude of which are written in terms of the poincare map of for suitable isolated but degenerate this enables us to obtain complete asymptotic expansions applying already known results about asymptotics of oscillatory integrals as an application we obtain lower bounds on the resonances close to the real axis for the laplace beltrami operator in r n n 2 for suitable compactly supported perturbations of the euclidean metric

reduced weyl asymptotics for pseudodifferential operators on bounded domains ii the compact group case abstract let g o n be a compact group of isometries acting on n dimensional euclidean space r n and x a bounded domain in r n which is transformed into itself under the action of g consider a symmetric classical pseudodifferential operator a 0 in l 2 r n that commutes with the regular representation of g and assume that it is elliptic on x we show that the spectrum of the friedrichs extension a of the operator res a 0 ext c c x l 2 x is discrete and using the method of the stationary phase we derive asymptotics for the number n of eigenvalues of a equal or less than and with eigenfunctions in the isotypic component of l 2 x as giving also an estimate for the remainder term for singular group actions since the considered critical set is a singular variety we recur to partial desingularization in order to apply the stationary phase theorem

a semi classical trace formula at a non degenerate critical level abstract we study the semi classical trace formula at a critical energy level for a h pseudodifferential operator whose principal symbol has a unique non degenerate critical point for that energy this leads to the study of hamiltonian systems near equilibrium and near the non zero periods of the linearized flow the contributions of these periods to the trace formula are expressed in terms of degenerate oscillatory integrals the new results obtained are formulated in terms of the geometry of the energy surface and the classical dynamics on this surface

the semi classical trace formula and propagation of wave packets abstract we study spectral and propagation properties of operators of the form s h n j 0 h j p j where j p j is a differential operator of order j on a manifold m asymptotically as h 0 the estimates are in terms of the flow t of the classical hamiltonian h x p n j 0 p j x p on t m where p j is the principal symbol of p j we present two sets of results i the semiclassical trace formula on the asymptotic behavior of eigenvalues and eigenfunctions of s h in terms of periodic trajectors of h ii associated to certain isotropic submanifolds t m we define families of functions h and prove that t exp iths is a family of the same kind associated to t

the radiation condition at infinity for the high frequency helmholtz equation with source term a wave packet approach abstract we consider the high frequency helmholtz equation with a given source term and a small absorption parameter 0 the high frequency or semi classical parameter is 0 we let and go to zero simultaneously we assume that the zero energy is non trapping for the underlying classical flow we also assume that the classical trajectories starting from the origin satisfy a transversality condition a generic assumption under these assumptions we prove that the solution u radiates in the outgoing direction uniformly in in particular the function u when conveniently rescaled at the scale close to the origin is shown to converge towards the outgoing solution of the helmholtz equation with coefficients frozen at the origin this provides a uniform version in of the limiting absorption principle writing the resolvent of the helmholtz equation as the integral in time of the associated semi classical schrodinger propagator our analysis relies on the following tools i for very large times we prove and use a uniform version of the egorov theorem to estimate the time integral ii for moderate times we prove a uniform dispersive estimate that relies on a wave packet approach together with the above mentioned transversality condition iii for small times we prove that the semi classical schrodinger operator with variable coefficients has the same dispersive properties as in the constant coefficients case uniformly in

traces and determinants of strongly stochastic operators periodic orbit theory allows calculations of long time properties of chaotic systems from traces dynamical zeta functions and spectral determinants of deterministic evolution operators which are in turn evaluated in terms of periodic orbits for the case of stochastic dynamics a direct numerical evaluation of the trace of an evolution operator is possible as a multidimensional integral techniques for evaluating such path integrals are discussed using as an example the logistic map f x lambdax 1 x with moderate to strong additive gaussian noise rapid convergence is demonstrated for all values of lambda with strong noise as well as at fixed lambda 5 for all noise levels

new code upper bounds from the terwilliger algebra and semidefinite programming we give a new upper bound on the maximum size a n d of a binary code of word length n and minimum distance at least d it is based on block diagonalizing the terwilliger algebra of the hamming cube the bound strengthens the delsarte bound and can be calculated with semidefinite programming in time bounded by a polynomial in n we show that it improves a number of known upper bounds for concrete values of n and d from this we also derive a new upper bound on the maximum size a n d w of a binary code of word length n minimum distance at least d and constant weight w again strengthening the delsarte bound and yielding several improved upper bounds for concrete values of n d and w

generic diffeomorphisms with superexponential growth of number of periodic orbits let m be a smooth compact manifold of dimension at least 2 and diff r m be the space of c r smooth diffeomorphisms of m associate to each diffeomorphism f 2 diff r m the sequence pn f of the number of isolated periodic points for f of period n in this paper we exhibit an open set n in the space of diffeomorphisms diff r m such for a baire generic diffeomorphism f 2 n the number of periodic points pnf grows with a period n faster than any following sequence of numbers fangn2zc along a subsequence i e pn f a ni for some ni 1 with i 1 in the cases of surface diffeomorphisms i e dim m d 2 an open set n with a supergrowth of the number of periodic points is a newhouse domain a proof of the man result is based on the gontchenko shilnikov turaev theorem gst a complete proof of that theorem is also presented

quantum ergodicity of boundary values of eigenfunctions suppose that n is a bounded piecewise smooth domain we prove that the boundary values cauchy data of eigenfunctions of the laplacian on with various boundary conditions are quantum ergodic if the classical billiard map on the ball bundle b is ergodic our proof is based on the classical observation that the boundary values of an interior eigenfunction 2 is an eigenfunction of an operator fh on the boundary of with h 1 in the case of the neumann boundary condition fh is the boundary integral operator induced by the double layer potential we show that fh is a semiclassical fourier integral operator quantizing the billiard map plus a small remainder the quantum dynamics defined by fh can be exploited on the boundary much as the quantum dynamics generated by the wave group were exploited in the interior of domains with corners and ergodic billiards in the work of zelditch zworski 1996 novelties include the facts that fh is not unitary and consequently the boundary values are equidistributed by measures which are not invariant under and which depend on the boundary conditions ergodicity of boundary values of eigenfunctions on domains with ergodic billiards was conjectured by s ozawa 1993 and was almost simultaneously proved by gerard leichtnam 1993 in the case of convex c1 1 domains with continuous tangent planes and with dirichlet boundary conditions our methods seem to be quite different motivation to study piecewise smooth domains comes from the fact that almost all known ergodic domains are of this form

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

contributions of non extremum critical points to the semi classical trace formula we study the semi classical trace formula at a critical energy level for a h pseudo differential operator on rn whose principal symbol has a totally degenerate critical point for that energy we compute the contribution to the trace formula of isolated non extremum critical points under a condition of real principal type the new contribution to the trace formula is valid for all time in a compact subset of r but the result is modest since we have restrictions on the dimension

recovering the hamiltonian from spectral data we show that the contributions to the gutzwiller formula with observable associated to the iterates of a given elliptic nondegenerate periodic trajectory gamma and to certain families of observables localized near gamma determine the quantum hamiltonian in a formal neighborhood of the trajectory gamma that is the full taylor expansion of its total symbol near gamma we also treat the bottom of a well case both for general and schrodinger operators

on tangential cohomology of riemannian foliations we prove kiinneth formula poincare duality hopf formula and index theorem for tangentially oriented compact riemannian foliations

convolutions of semi classical spectral distributions and periodic orbit theory abstract we study the convolution of semi classical spectral distributions associated to h pseudodifferential operators on r n under standard assumptions the micro support of this object can be characterized via families of periodic orbits correlated simultaneously by energy and periods when all the orbits are non degenerate the convolution admits as h tends to 0 an explicit asymptotic expansion in term of the respective dynamical systems in this setting this result validates the theory of orbits pairs used by physicists in quantum chaos some new contributions related to the crossing of the period functions are also analyzed

calcul fonctionnel par la transformation de mellin et operateurs admissibles mellin s transform is used to establish a functional calculus of a class of pseudodifferential operators depending on a small parameter h 0 we apply for exeample this result to prove the semi classical behaviour of the discrete spectrum of schrodinger operators h2 v and of dirac operators h j 13 jdj 4 v

trace formulas for stochastic evolution operators weak noise perturbation theory periodic orbit theory is all effective tool for the analysis of classical and quantum chaotic systems in this paper we extend this approach to stochastic systems in particular to mappings with additive noise the theory is cast in the standard field theoretic formalism and weak noise perturbation theory written in terms of feynman diagrams the result is a stochastic analog of the next to leading h corrections to the gutzwiller trace formula with long time averages calculated from periodic orbits of the deterministic system the perturbative corrections are computed analytically and tested numerically on a simple 1 dimensional system

chaotic field theory a sketch abstract spatio temporally chaotic dynamics of a classical field can be described by means of an infinite hierarchy of its unstable spatio temporally periodic solutions the periodic orbit theory yields the global averages characterizing the chaotic dynamics as well as the starting semiclassical approximation to the quantum theory new methods for computing corrections to the semiclassical approximation are developed in particular a nonlinear field transformation yields the perturbative corrections in a form more compact than the feynman diagram expansions

inference from stratified samples properties of the linearization jackknife and balanced repeated replication methods the asymptotic normality of both linear and nonlinear statistics and the consistency of the variance estimators obtained using the linearization jackknife and balanced repeated replication brr methods in stratified samples are established the results are obtained as l oa within the context of a sequence of finite populations jil with l strata in ji l and are valid for any stratified multistage design in which the primary sampling units psu s are selected with replacement and in which independent subsamples are taken within those psu s selected more than once in addition some exact analytical results on the bias and stability of these alternative variance estimators in the case of ratio estimation are obtained for small l under a general linear regression model

finite dimensional approximations to wiener measure and path integral formulas on manifolds abstract certain natural geometric approximation schemes are developed for wiener measure on a compact riemannian manifold these approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi group on riemannian manifolds the path space is approximated by finite dimensional manifolds h p m consisting of piecewise geodesic paths adapted to partitions p of 0 1 the finite dimensional manifolds h p m carry both an h 1 and a l 2 type riemannian structures g 1 p and g 0 p respectively it is proved that 1 z i p e 1 2 e d vol g i p i d as mesh p 0 where e is the energy of the piecewise geodesic path h p m and for i 0 and 1 z i p is a normalization constant vol g i p is the riemannian volume form relative to g i p and is wiener measure on paths on m here 1 1 and 0 exp 1 6 1 0 scal s ds where scal is the scalar curvature of m these results are also shown to imply the well known integration by parts formula for the wiener measure

hamiltonian evolution of monokinetic measures with rough momentum profile consider a monokinetic probability measure on the phase space bf r n x times bf r n xi i e mu rm in rho rm in x delta xi u rm in x where uin is a vector field on rn and in a probability density on rn let t be a hamiltonian flow on rn rn in this paper we study the structure of the transported measure mu t phi t mu rm in and of its integral in the variable denoted t in particular we give estimates on the number of folds in phi t rm graph of u rm in on which t is concentrated we explain how our results can be applied to investigate the classical limit of the schrodinger equation by using the formalism of wigner measures our formalism includes initial momentum profiles uin with much lower regularity than required by the wkb method finally we discuss a few examples showing that our results are sharp

infinite kneading matrices and weighted zeta functions of interval maps abstract we consider a piecewise continuous piecewise monotone interval map and a weight of bounded variation constant on homtervals and continuous at periodic points of the map with these data we associate a sequence of weighted milnor thurston kneading matrices converging to a countable matrix with coefficients analytic functions we show that the determinants of these matrices converge to the inverse of the correspondingly weighted zeta function for the map as a corollary we obtain convergence of the discrete spectrum of the perron frobenius operators of piecewise linear approximations of markovian piecewise expanding and piecewise c 1 bv interval maps

semidefinite programming bounds for constant weight codes for nonnegative integers n d and w let a n d w be the maximum size of a code c subseteq mathbb f 2 n with a constant weight w and minimum distance at least d we consider two semidefinite programs based on quadruples of code words that yield several new upper bounds on a n d w the new upper bounds imply that a 22 8 10 616 and a 22 8 11 672 lower bounds on a 22 8 10 and a 22 8 11 are obtained from the n d 22 7 shortened golay code of size 2048 it can be concluded that the shortened golay code is a union of constant weight w codes of sizes a 22 8 w

the analysis of partial differential operators i hormander springer 1985 vol 1 391 pp vol 2 390 pp vol 3 525 pp vol 4 351 pp

anosov flows and dynamical zeta functions we study the ruelle and selberg zeta functions for c r anosov ows r 2 on a compact smooth manifold we prove several results the most remarkable being a for c 1 ows the zeta function is meromorphic on the entire complex plane b for contact ows satisfying a bunching condition e g geodesic ows on manifolds of negative curvature better than 1 9 pinched the zeta function has a pole at the topological entropy and is analytic in a strip to its left c under the same hypotheses as in b we obtain sharp results on the number of periodic orbits our arguments are based on the study of the spectral properties of a transfer operator acting on suitable banach spaces of anisotropic currents

dynamical entropy for systems with stochastic perturbation dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively since for such systems the kolmogorov sinai ks entropy diverges if the diameter of the partition tends to zero we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself we show that this quantity is finite and non negative and we call it the dynamical entropy of the noisy system in the weak noise limit this quantity is conjectured to tend to the ks entropy of the deterministic system in particular we consider one dimensional systems with noise described by a finite dimensional kernel for which the frobenius perron operator can be represented by a finite matrix

weak noise approach to the logistic map using a nonperturbative weak noise approach we investigate the interference of noise and chaos in simple 1d maps we replace the noise driven 1d map by an area preserving 2d map modelling the poincare sections of a conserved dynamical system with unbounded energy manifolds we analyze the properties of the 2d map and draw conclusions concerning the interference of noise on the nonlinear time evolution we apply this technique to the standard period doubling sequence in the logistic map from the 2d area preserving analogue we in addition to the usual period doubling sequence obtain a series of period doubled cycles which are elliptic in nature these cycles are spinning off the real axis at parameters values corresponding to the standard period doubling events

spectral fluctuations of schrodinger operators generated by critical points of the potential starting from the spectrum of schrodinger operators on mathbb r n we propose a method to detect critical points of the potential we argue semi classically on the basis of a mathematically rigorous version of gutzwiller s trace formula which expresses spectral statistics in term of classical orbits a critical point of the potential with zero momentum is an equilibrium of the flow and generates certain singularities in the spectrum via sharp spectral estimates this fluctuation indicates the presence of a critical point and allows to reconstruct partially the local shape of the potential some generalizations of this approach are also proposed

semiclassical sum rules and generalized coherent states various semiclassical sum rules are obtained for matrix elements of smooth observables using semiclassical estimates of time evolved coherent states together with stationary phase theorems

a semi classical trace formula for schrodinger operators in the case of a critical energy level let 2 2 v x be a schrodinger operator on rn with smooth potentialv x as x the spectrum of is discrete and one can study the asymptotic of the smoothed spectral density e k ek e as 0 here ek k nis the spectrum of and c 0 r we shall investigate the case whereeis a critical value of the symbolhof and extending the work of brummelhuis paul and uribe in 3 we will prove the existence of a full asymptotic expansion for in terms of and ln and compute the leading coefficient we will consider new weyl type estimates for the counting function nec k n ek e

asymptotic expansions for the escape rate of stochastically perturbed unimodal maps abstract the escape rate of a stochastic dynamical system can be found as an expansion in powers of the noise strength in the previous work the coefficients of such an expansion for a one dimensional map were fitted to a general form containing a few parameters these parameters were found to be related to the fractal structure of the repeller of the system the parameter the noise dimension remains to be interpreted this report presents new data for showing that the relation to the dimensions is more complicated than predicted in the earlier work and oscillates as a function of the map parameter in contrast to other dimension like quantities

algebras of semiclassical pseudodifferential operators associated with zoll type domains in cotangent bundles abstract we consider domains in cotangent bundles with the property that the null foliation of their boundary is fibrating and the leaves satisfy a bohr sommerfeld condition for example the unit disk bundle of a zoll metric given such a domain we construct an algebra of associated semiclassical pseudodifferential operators with singular symbols the schwartz kernels of the operators have frequency set contained in the union of the diagonal and the flow out of the null foliation of the boundary of the domain we develop a symbolic calculus prove the existence of projectors under a mild additional assumption whose range can be thought of as quantizing the domain give a symbolic proof of a szego limit theorem and study associated propagators

semiclassical spectral estimates for schrodinger operators at a critical energy level case of a degenerate minimum of the potential abstract we study the semi classical trace formula at a critical energy level for a schrodinger operator on r n we assume here that the potential has a totally degenerate critical point associated to a local minimum the main result which computes the contribution of this equilibrium is valid for all time in a compact and establishes the existence of a total asymptotic expansion whose top order coefficient depends only on the germ of the potential at the critical point

noisy one dimensional maps near a crisis i weak gaussian white and colored noise we study one dimensional single humped maps near the boundary crisis at fully developed chaos in the presence of additive weak gaussian white noise by means of a new perturbation like method the quasi invariant density is calculated from the invariant density at the crisis in the absence of noise in the precritical regime where the deterministic map may show periodic windows a necessary and sufficient condition for the validity of this method is derived from the quasi invariant density we determine the escape rate which has the form of a scaling law and compares excellently with results from numerical simulations we find that deterministic transient chaos is stabilized by weak noise whenever the maximum of the map is of orderz 1 finally we extend our method to more general maps near a boundary crisis and to multiplicative as well as colored weak gaussian noise within this extended class of noises and for single humped maps with any fixed orderz 0 of the maximum in the scaling law for the escape rate both the critical exponents and the scaling function are universal

jackknifing in the presence of inhomogeneity under the classical assumption that data are a random sample from a distribution with cumulative distribution function f the jackknife generally yields bias reduction an asymptotically pivotal statistic and a variance estimator for an estimator of an unknown parameter f in this article the classical assumption is relaxed to allow for inhomogeneous subpopulations the jackknife is been to account for these inhomogeneities automatically and so is valid in a class of problems much larger than that for which it was originally intended data from experiments to determine the acceleration of gravity at washington d c are analyzed a family of weighted mean estimators is considered and recommendations are made regarding which estimators yield both valid and efficient jackknife based inferences

heteroscedasticity robustness of jackknife variance estimators in linear models on etudie l absence de biais et la consistance asymptotiques de trois types d estimateurs de variance du jackknife en presence d heteroscedasticite de la variance de l erreur dans les modeles lineaires

noisy one dimensional maps near a crisis ii general uncorrelated weak noise the escape rate for one dimensional noisy maps near a crisis is investigated a previously introduced perturbation theory is extended to very general kinds of weak uncorrelated noise including multiplicative white noise as a special case for single humped maps near the boundary crisis at fully developed chaos an asymptotically exact scaling law for the rate is derived it predicts that transient chaos is stabilized by basically any noise of appropriate strength provided the maximum of the map is of sufficiently large order a simple heuristic explanation of this effect is given the escape rate is discussed in detail for noise distributions of levy dichotomous and exponential type in the latter case the rate is dominated by an exponentially leading arrhenius factor in the deep precritical regime however the preexponential factor may still depend more strongly than any power law on the noise strength

an approximation to wiener measure and quantization of the hamiltonian on manifolds with non positive sectional curvature this paper gives a rigorous interpretation of a feynman path integral on a riemannian manifold m with non positive sectional curvature an l2 riemannian metric gp is given on the space of piecewise geodesic paths hp m adapted to the partition p of 0 1 whence a finite dimensional approximation of wiener measure is developed it is proved that as mesh p 0 the approximate wiener measure converges in an l1 sense to the measure exp 2 3203 01scal s ds d on the wiener space w m with wiener measure this gives a possible prescription for the path integral representation of the quantized hamiltonian as well as yielding such a result for the natural geometric approximation schemes originating in andersson and driver 1999 3 and followed by lim 2007 34

some remarks about semiclassical trace invariants and quantum normal forms in this paper we explore the connection between semi classical and quantum birkhoff canonical forms bcf for schrodinger operators in particular we give a non symbolic operator theoretic derivation of the quantum birkhoff canonical form and provide an explicit recipe for expressing the quantum bcf in terms of the semi classical bcf

equivariant spectral asymptotics for h pseudodifferential operators we prove equivariant spectral asymptotics for h pseudodifferential operators for compact orthogonal group actions generalizing results of el houakmi and helffer comportement semi classique en presence de symetries action d un groupe de lie compact asymp anal 5 2 91 113 1991 and cassanas reduced gutzwiller formula with symmetry case of a lie group j math pures appl 85 6 719 742 2006 using recent results for certain oscillatory integrals with singular critical sets p ramacher singular equivariant asymptotics and weyl s law on the distribution of eigenvalues of an invariant elliptic operator j reine angew math crelles j to be published we can deduce a weak equivariant weyl law furthermore we can prove a complete asymptotic expansion for the gutzwiller trace formula without any additional condition on the group action by a suitable generalization of the dynamical assumptions on the hamilton flow

periodic orbit sum rules for billiards accelerating cycle expansions we show that the periodic orbit sums for two dimensional billiards satisfy an infinity of exact sum rules we demonstrate their utility by using the flow conservation sum rule to accelerate the convergence of cycle expansions for the overlapping three disc billiard the effectiveness of the approach is studied by applying the method on averages known explicitly by other sum rules the method is then applied to the lyapunov exponent

stochastic analysis on the path space of a riemannian manifold i markovian stochastic calculus abstract on the brownian flow of a compact riemannian manifold an intrinsic stochastic calculus is defined this calculus is markovian the stochastic calculus of variation on the bundle of orthonormal frames appears only as an intermediate tool the bismut connection plays a paramount role in the final construction

constructive field theory and applications perspectives and open problems in this paper we review many interesting open problems in mathematical physics which may be attacked with the help of tools from constructive field theory they could give work for future mathematical physicists trained with constructive methods well into the 21st century

semidefinite code bounds based on quadruple distances let a n d be the maximum number of 0 1 words of length n any two having hamming distance at least d it is proved that a 20 8 256 which implies that the quadruply shortened golay code is optimal moreover it is shown that a 18 6 673 a 19 6 1237 a 20 6 2279 a 23 6 13674 a 19 8 135 a 25 8 5421 a 26 8 9275 a 27 8 17099 a 21 10 47 a 22 10 84 a 24 10 268 a 25 10 466 a 26 10 836 a 27 10 1585 a 28 10 2817 a 25 12 55 and a 26 12 96 the method is based on the positive semidefiniteness of matrices derived from quadruples of words this can be put as constraint in a semidefinite program whose optimum value is an upper bound for a n d the order of the matrices involved is huge however the semidefinite program is highly symmetric by which its feasible region can be restricted to the algebra of matrices invariant under this symmetry by block diagonalizing this algebra the order of the matrices will be reduced so as to make the program solvable with semidefinite programming software in the above range of values of n and d

the dimension of the leafwise reduced cohomology geometric conditions are given so that the leafwise reduced cohomology is of infinite dimension especially for foliations with dense leaves on closed manifolds the main new definition involved is the intersection number of subfoliations with appropriate coefficients the leafwise reduced cohomology is also described for homogeneous foliations with dense leaves on closed nilmanifolds

transformations and the jackknife summary whether to jackknife a parameter estimate or some function of it has in the past relied on recommendations based on folklore rather than statistical analyses this paper puts the jackknifed estimator into the context of finding a least squares estimator from a model involving a location parameter with additive pseudo errors obtaining and interpreting confidence intervals from the partitioning of the total sum of squares into residual sum of squares plus a sum of squares due to a hypothesized value of the parameter requires the error variance to be constant this in turn can equally be interpreted as requiring the variance of the original estimator to be constant therefore using this criterion of jackknifing pivotal quantities it is possible in principle to find a prejackknifing transformation for any general problem

weighted trace formula near a hyperbolic trajectory and complex orbits in this paper we consider a weighted trace formula for schrodinger operators more precisely let j and ej denote the eigenfunctions and eigenvalues of a schrodinger type operator h with a discrete spectrum let x be a coherent state centered at a point x of a hyperbolic closed orbit we show that as 0 the leading term of j ej e x j 2 can be expressed in terms of the analytic continuation on the upper and lower half planes of the positive and negative frequencies part of the result is also related to complex trajectories surrounding

inverse spectral problems for schrodinger operators in this article we find some explicit formulas for the semi classical wave invariants at the bottom of the well of a schrodinger operator as an application of these new formulas for the wave invariants we improve the inverse spectral results proved by guillemin and uribe in gu they proved that under some symmetry assumptions on the potential v x the taylor expansion of v x near a non degenerate global minimum can be recovered from the knowledge of the low lying eigenvalues of the associated schrodinger operator in mathbb r n we prove similar inverse spectral results using fewer symmetry assumptions we also show that in dimension 1 no symmetry assumption is needed to recover the taylor coefficients of v x

on the small scale mass concentration of modes let be commuting jointly elliptic pseudodifferential operators on a compact manifold x of dimension n d suppose is the limit set of the bicharacteristic flow of the classical hamiltonian p 1 restricted to the variety we discuss the corresponding concentration of mass as for a subsequence of joint eigenfunctions of the p j s with eigenvalues sufficiently close to

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

gauge invariant quantization on riemannian manifolds for every pointwise polynomial function on each fiber of the cotangent bundle of a riemannian manifold m a family of diffential operators is given which acts on the space of smooth sections of a vector bundle on m such a correspondence may be considered as a rule to quantize classical systems moving in a riemannian manifold or in a gauge field some applications of our construction are also given in this paper

a dynamical zeta function for pseudo riemannian foliations we investigate a generalization of geodesic random walks to pseudo riemannian foliations the main application we have in mind is to consider the logarithm of the associated zeta function as grand canonical partition function in a theory unifying aspects of general relativity quantum mechanics and dynamical systems

a proof of the gutzwiller semiclassical trace formula using coherent states decomposition the gutzwiller trace formula links the eigenvalues of the schrodinger operator as planck s constant goes to zero the semiclassical regime with the closed orbits of the corresponding classical mechanical system gutzwiller gave a heuristic proof of this trace formula using the feynman integral representation for the propagator of later using the theory of fourier integral operators mathematicians gave rigorous proofs of the formula in various settings here we show how the use of coherent states allows us to give a simple and direct proof

stochastic jacobi fields and vector fields induced by varying area on path spaces we study two classes of vector fields on the path space over a closed manifold with a wiener riemannian measure by adopting the viewpoint of yang mills field theory we study a vector field defined by varying a metric connection we prove that the vector field obtained in this way satisfies a jacobi field equation which is different from that of classical one by taking in account that a brownian motion is invariant under the orthogonal group action so that it is a geometric vector field on the space of continuous paths and induces a quasi invariant solution flow on the path space the second object of this paper is vector fields obtained by varying area here we follow the idea that a continuous semimartingale is indeed a rough path consisting of not only the path in the classical sense but also its levy area we prove that the vector field obtained by parallel translating a curve in the initial tangent space via a connection is just the vector field generated by translating the path along a direction in the cameron martin space in the malliavin calculus sense and at the same time changing its levy area in an appropriate way this leads to a new derivation of the integration by parts formula on the path space

entire fredholm determinants for evaluation of semiclassical and thermodynamical spectra proofs that fredholm determinants of transfer operators for hyperbolic flows are entire can be extended to a large new class of multiplicative evolution operators we construct such operators both for the gutzwiller semiclassical quantum mechanics and for classical thermodynamic formalism and introduce a new functional determinant which is expected to be entire for axiom a flows and whose zeros coincide with the zeros of the gutzwiller voros zeta function

a new table of constant weight codes a table of binary constant weight codes of length nl28 is presented explicit constructions are given for most of the 600 codes in the table the majority of these codes are new the known techniques for constructing constant weight codes are surveyed and a table of unrestricted binary codes of length nl28 is given

a semiclassical sum rule for matrix elements of classically chaotic systems in the semiclassical limit the sum s e delta e sigma nm mod anm mod 2 delta e 1 2 en em delta delta e en em of matrix elements of an arbitrary operator a can be related to the classical correlation function of the weyl symbol a q p of a ca e t integral d alpha delta e h alpha a alpha a alpha t s e delta e is proportional to the fourier transform of ca e t over t plus a set of correction terms associated with periodic trajectories in phase space if the system has a chaotic classical limit the matrix elements are independently gaussian distributed with mean value zero and s e delta e gives the variance of this distribution

a semi classical trace formula at a totally degenerate critical level we study the semi classical trace formula at a critical energy level for an h pseudo differential operator on open image in new window n whose principal symbol has a totally degenerate critical point for that energy this problem is studied for a large time behavior and under the hypothesis that the principal symbol of the operator has a local extremum at the critical point

semi classical spectral estimates for schrodinger operators at a critical level case of a degenerate maximum of the potential we study the semi classical trace formula at a critical energy level for a schrodinger operator on r n we assume here that the potential has a totally degenerate crit ical point associated to a local maximum the main result which establishes the contribution of the associated equilibrium in the trace formula is valid for all time in a compact subset of r and includes the singularity in t 0 for these new contributions the asymptotic expansion involves the logarithm of the parameter h depending on an explicit arithmetic condition on the dimension and the order of the critical point this logarithmic contribution can appear in the leading term

sharp semiclassical estimates for the number of eigenvalues below a totally degenerate critical level we consider the semiclassical asymptotic behaviour of the number of eigenvalues smaller than e for elliptic operators in l2 rd we describe a method of obtaining remainder estimates related to the volume of the region of the phase space in which the principal symbol takes values belonging to the intervals e e h where e is close to e if the volume of this region is o h then we obtain remainder estimates o h1 d with no assumptions on the hessian of the principal symbol at the critical level moreover we do not assume that the coefficients are smooth all results hold if second order derivatives of coefficients are holder continuous

trace formula for noisy flows a trace formula is derived for the fokker planck equation associated with ito stochastic differential equations describing noisy time continuous nonlinear dynamical systems in the weak noise limit the trace formula provides estimations of the eigenvalues of the fokker planck operator on the basis of the pollicott ruelle resonances of the noiseless deterministic system which is assumed to be non bifurcating at first order in the noise amplitude the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo energy of the onsager machlup freidlin wentzell scheme

semiclassical form factor for spectral and matrix element fluctuations of multidimensional chaotic systems we present a semiclassical calculation of the generalized form factor kab tau which characterizes the fluctuations of matrix elements of the operators a and b in the eigenbasis of the hamiltonian of a chaotic system our approach is based on some recently developed techniques for the spectral form factor of systems with hyperbolic and ergodic underlying classical dynamics and f 2 degrees of freedom that allow us to go beyond the diagonal approximation first we extend these techniques to systems with f 2 then we use these results to calculate kab tau we show that the dependence on the rescaled time tau time in units of the heisenberg time is universal for both the spectral and the generalized form factor furthermore we derive a relation between kab tau and the classical time correlation function of the weyl symbols of a and b

circular symmetry and the trace formula let m be a compact n dimensional manifold without boundary and let p be a positive first order elliptic self adjoint classical pseudodifferential operator on lz m the hilbert space of half densities on m the spectrum of p is discrete and accumulates only at we denote the eigenvalues with multiplicities by 2o 21 2 recall that the singularities of the inverse fourier transform of the spectral measure