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

resonances and spectral shift function for a magnetic schrodinger operator we consider the three dimensional schrodinger operator h0 with a constant magnetic field and subject to an electric potential v0 depending only on the variable along the magnetic field x3 the operator h0 has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum we perturb h0 by smooth scalar potentials v o x1 x2 x3 2 1 we assume also that v and v0 have an analytic continuation in the magnetic field direction in a complex sector outside a compact set we define the resonances of h h0 v as the eigenvalues of the nonself adjoint operator obtained from h by analytic distortions of rx3 we study their distribution near any fixed real eigenvalue of h0 2bq for q n in a ring centered at 2bq with radii r 2r we establish an upper bound as r tends to 0 of the number of resonances this upper bound depends on the decay of v at infinity only in the directions x1 x2 finally we deduce a representation of the derivative of the spectral shift funct

multifractal analysis of conformal axiom a flows we develop the multifractal analysis of conformal axiom a flows this includes the study of the hausdorff dimension of basic sets of the flow the description of the dimension spectra for pointwise dimension and for lyapunov exponents and the multifractal decomposition associated with these spectra the main tool of study is the thermodynamic formalism for hyperbolic flows by bowen and ruelle examples include suspensions over axiom a conformal diffeomorphisms anosov flows and in particular geodesic flows on compact smooth surfaces of negative curvature

threshold for monotone symmetric properties through a logarithmic sobolev inequality threshold phenomena are investigated using a general approach following talagrand ann probab 22 1994 1576 1587 and friedgut and kalai proc amer math soc 12 1999 1017 10541 the general upper bound for the threshold width of symmetric monotone properties is improved this follows from a new lower bound on the maximal influence of a variable on a boolean function the method of proof is based on a well known logarithmic sobolev inequality on 0 1 n this new bound is shown to be asymptotically optimal

scattering theory and the trace of the wave group in 9 lax and phillips discuss the analogue for classical scattering theories in odd space dimensions of the poisson summation formula as generalized to compact manifolds by chazarain 3 colin de verdi e 141 and duistermaat and guillemin 5 and further generalized to manifolds with boundary in 1 71 however in 9 this trace formula is only proved for comparatively large values of the time variable see also ralston 14 1 bardos el al 2 this note is devoted to the proof of the same formula for all positive time for the case of scattering by a smooth compactly supported potential if n a is the number of poles a 3 then for every e 0 there exists c 0 such that

sharp upper bounds on the number of resonances near the real axis for trapping systems we study resonances near the real axis jimz jd o h n n 1 and the corresponding resonant states for semiclassical long range operators p h without a priori assumptions on the distribution or on the multiplicities of the resonances we show that the truncated resonant states form a family of quasimode states for p h stable under small perturbations as a consequence they form also a family of quasimode states for any suitably defined selfadjoint reference operator p h therefore those resonances are perturbed eigenvalues of p h next we show that the semiclassical wave front set of the resonant states is contained in the set of trapped directions t we construct a suitable reference operator from p h by imposing a microlocal barrier outside t to show that the counting function for those resonances admits an upper bound of weyl s type connected with the measure of t we give an example of system for which this bound is optimal and also prove similar bound in case of classical scattering by obstacle this paper is devoted to a detailed study of the behavior of the resonances and resonant states near the real axis we work mainly in the semi classical setting but most results can be easily translated into the classical one by resonances near the real axis we mean resonances in a box h d a0 b0 ci s h 0 where 0 s h d o h k k 1 such resonances may exist only for trapping geometries we accept the convention here that resonances lie in the lower half plane

note on the distribution of poles of the scattering matrix abstract trapping obstacles having at least one isolated multiple reflecting trapping ray are considered it is proved that for every e 0 there are poles j of the scattering matrix such that 0 im j log j

asymptotics of resonances for a schrodinger operator with vector values abstract we show that the resonance counting function for a schrodinger operator in dimension one has an asymptotic expansion and calculate an explicit expression for the leading term in some situations

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

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

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

sharp polynomial bounds on the number of scattering poles of radial potentials abstract it is shown that for scattering by a radially symmetric potential in r n n odd the number of poles in a disk of radius r satisfies an estimate n r c n r 1 n this bound is sharp as shown by the special case of potentials nonvanishing at the boundary where n r k n a n r n 1 o 1 a being the diameter of the support

a lower bound on the variance of conductance in random resistor networks we study the conductance of random resistor networks ind 2 dimensions it is shown under some technical assumptions that if a network exhibits a nonzero conductivity in the infinite volume limit then the variance of a finite volume conductance grows at least like the volume

existence of resonances in magnetic scattering the schrodinger operator with a compactly supported magnetic field is shown to produce infinitely many resonances in any odd dimension 3 the proof is based on the poisson formula for resonances and properties of the magnetic heat invariants

submean variance bound for effective resistance of random electric networks we study a model of random electric networks with bernoulli resistances in the case of the lattice mathbb z 2 we show that the point to point effective resistance between 0 and a vertex v has a variance of order at most rm log v frac 2 3 whereas its expected value is of order log v when v goes to infinity when d 2 expectation and variance are of the same order similar results are obtained in the context of p resistance the proofs rely on a modified poincare inequality due to falik and samorodnitsky 7

resonances for non trapping time periodic perturbations for time periodic perturbations of the wave equation in given by a potential q t x we obtain an upper bound of the number of the resonances we establish for m n large enough a trace formula relating the iterations of the monodromy operator u mt 0 t 0 and the sum jzmj of all resonances counted with their multiplicities

hitting and commute times in large random neighborhood graphs in machine learning a popular tool to analyze the structure of graphs is the hitting time and the commute distance resistance distance for two vertices u and v the hitting time huv is the expected time it takes a random walk to travel from u to v the commute distance is its symmetrized version cuv huv hvu in our paper we study the behavior of hitting times and commute distances when the number n of vertices in the graph tends to infinity we focus on random geometric graphs e graphs knn graphs and gaussian similarity graphs but our results also extend to graphs with a given expected degree distribution or erdos renyi graphs with planted partitions we prove that in these graph families the suitably rescaled hitting time huv converges to 1 dv and the rescaled commute time to 1 du 1 dv where du and dv denote the degrees of vertices u and v in these cases hitting and commute times do not provide information about the structure of the graph and their use is discouraged in many machine learning applications

lower bounds on the number of scattering poles ii abstract we obtain lower bounds on the number of scattering poles for a class of abstract compactly supported perturbations of the laplacian in r n n odd they are applied to estimate the number of resonances for obstacle scattering and for hypoelliptic compactly supported perturbations of the laplacian the proof involves a development of lax phillips theory in a generalized setting a poisson formula for scattering poles and some simple tauberian arguments

several complex variables and the distribution of resonances in potential scattering we study resonances associated to schrodinger operators with compactly supported potentials on d d 3 odd we consider potentials depending holomorphically on a parameter open image in new window for certain such families for all z except those in a pluripolar set the associated resonance counting function has order of growth d

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

an inverse problem for the wave equation with first order perturbation pour l equation d onde sur r n avec perturbation du potentiel on montre que l on peut retrouver le potentiel a partir du comportement asymptotique de l amplitude de diffusion on montre qu un resultat semblable est vrai quand on ajoute des termes d ordre 1 a l equation

semi classical asymptotics in solid state physics this article studies the schrodinger equation for an electron in a lattice of ions with an external magnetic field in a suitable physical scaling the ionic potential becomes rapidly oscillating and one can build asymptotic solutions for the limit of zero magnetic field by multiple scale methods from homogenization for the time dependent schrodinger equation this construction yields wave packets which follow the trajectories of the semiclassical model for the time independent equation one gets asymptotic eigenfunctions or quasimodes for the energy levels predicted by onsager s relation

transience recurrence and critical behavior for long range percolation we study the behavior of the random walk on the infinite cluster of independent long range percolation in dimensions d 1 2 where x and y are connected with probability we show that if d s 2d then there is no infinite cluster at criticality this result is extended to the free random cluster model a second corollary is that when d 2 and d s 2d we can erase all long enough bonds and still have an infinite cluster the proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks in particular we show that i i d conductances on a recurrent graph of bounded degree yield a recurrent electrical network

existence of resonances in three dimensions ifp is an elliptic self adjoint perturbation of the laplacian on 3 and the coefficients ofp decay super exponentially then we show thatp has infinitely many resonances the resonances are defined here as the poles of the meromorphic continuation of p 2 1

an efron stein inequality for nonsymmetric statistics on etablit l analogue d un resultat d efron et stein 1981 que l on demontre a l aide d une technique d espace de hilbert introduite par vitale 1984

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

neumann resonances in linear elasticity for an arbitrary body we study resonances scattering poles associated to the elasticity operator in the exterior of an arbitrary obstacle inr3 with neumann boundary conditions we prove that there exists a sequence of resonances tending rapidly to the real axis

spectral shift function and resonances for slowly varying perturbations of periodic schrodinger operators we study the spectral shift function s h and the resonances of the operator p h v x w hx here v is a periodic potential w a decreasing perturbation and h a small positive constant we give a representation of the derivative of s h related to the resonances of p h and we obtain a weyl type asymptotics of s h we establish an upper bound o h n 1 for the number of the resonances of p h lying in a disk of radius h

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

the diffusion limit for reversible jump processes on z d with ergodic random bond conductivities we consider a reversible jump process on d whose jump rates themselves are random we show mean square convergence of this process under diffusion scaling to a limiting brownian motion with a certain diffusion matrix characterizing effective conductivity

asymptotics on the number of scattering poles for degenerate perturbations of the laplacian abstract for a class of compactly supported hypoelliptic perturbations of the laplacian in r n n 3 odd we prove that an asymptotic on the number of the eigenvalues of the corresponding reference operator implies a similar asymptotic for the number of the scattering poles

distribution of scattering resonances for generic schrodinger operators abstract let v be the schrodinger operator acting on l 2 r d c with d 3 odd here v is a bounded real or complex valued function vanishing outside the closed ball of center 0 and radius a if v belongs to the class m a of potentials introduced by christiansen we show that when r the resonances of v scaled down by the factor r are asymptotically distributed with respect to an explicit probability distribution on the lower unit half disc of the complex plane the rate of convergence is also considered for subclasses of potentials

a central limit theorem for the effective conductance linear boundary data and small ellipticity contrasts given a resistor network on mathbb z d with nearest neighbor conductances the effective conductance in a finite set with a given boundary condition is the minimum of the dirichlet energy over functions with the prescribed boundary values for shift ergodic conductances linear dirichlet boundary conditions and square boxes the effective conductance scaled by the volume of the box converges to a deterministic limit as the box size tends to infinity here we prove that for i i d conductances with a small ellipticity contrast also a non degenerate central limit theorem holds the proof is based on the corrector method and the martingale central limit theorem a key integrability condition is furnished by the meyers estimate more general domains boundary conditions and ellipticity contrasts will be addressed in a subsequent paper

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

asymptotic number of scattering resonances for generic schrodinger operators let v be the schrodinger operator acting on l 2 mathbb r d mathbb c with d geq 3 odd here v is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a let n v r denote the number of resonances of v with modulus r we show that if the potential v is generic in a sense of pluripotential theory then n v r c d a dr d o r d 3 over 16 epsilon quad mbox as r to infty for any e 0 where c d is a dimensional constant

on russo s approximate zero one law consider the product measure mu p on 0 1 n when 0 operatorname resp 1 is given weight 1 p operatorname resp p consider a monotone subset a of 0 1 n we give a precise quantitative form to the following statement if a does not depend much on any given coordinate d mu p a dp is large thus in that case there is a threshold effect and mu p a jumps from near 0 to near 1 in a small interval

a mathematical approach to the effective hamiltonian in perturbed periodic problems we describe a rigorous mathematical reduction of the spectral study for a class of periodic problems with perturbations which gives a justification of the method of effective hamiltonians in solid state physics we study the partial differential operators of the formp p hy y d y a hy onr n whenh 0 is small enough wherep x y is elliptic periodic iny with respect to some lattice and admits smooth bounded coefficients in x y a x is a magnetic potential with bounded derivatives we show that the spectral study ofp near any fixed energy level can be reduced to the study of a finite system ofh pseudodifferential operatorse x hd x h acting on some hilbert space depending on we then apply it to the study of the schrodinger operator when the electric potential is periodic and to some quasiperiodic potentials with vanishing magnetic field

first passage percolation has sublinear distance variance let 0 1 the distance dist 0 from the origin to a vertex 2 has variance bounded by c log where c c a b d is a constant which may only depend on a b and d some related variants are also discussed

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

sharp upper bounds on the number of the scattering poles abstract we study the scattering poles of a compactly supported black box perturbations of the laplacian in r n n odd we prove a sharp upper bound of the counting function n r modulo o r n in terms of the counting function of the reference operator in the smallest ball around the black box in the most interesting cases we prove a bound of the type n r a n r n o r n with an explicit a n we prove that this bound is sharp in a few special spherically symmetric cases where the bound turns into an asymptotic formula

resonances in one dimension and fredholm determinants we discuss resonances for schrodinger operators in whole and half line problems one of our goals is to connect the fredholm determinant approach of froese to the fourier transform approach of zworski another is to prove a result on the number of antibound states namely in a half line problem there are an odd number of antibound states between any two bound states

a local trace formula for resonances of perturbed periodic schrodinger operators abstract we give a local trace formula for the pair p 1 h p 0 w hy p 0 where p 0 is a periodic schrodinger operator w is a decreasing perturbation and h is a small positive parameter we apply this result to establish the existence of h n resonances near some energy of p 0

asymptotic distribution of resonances in one dimension abstract we determine the leading asymptotics of the resonance counting function for a class of schrodinger operators in one dimension whose potentials may have non compact support

scattering frequencies and gevrey 3 singularities on etudie les frequences de diffusion k pour le laplacien dans la region r n o exterieur a un obstacle analytique compact on suppose que n est impair et qu on impose des conditions de dirichlet sur

sharp bounds on the number of scattering poles for perturbations of the laplacian sharp bounds on the numbern r of the scattering poles in the disc z r for a large class of compactly supported perturbations not necessarily selfadjoint of the laplacian in n n 3 odd are obtained in particular in the elliptic case the estimaten r cr n c is proved

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