spectral shift function for trapping energies in the semiclassical limit semiclassical asymptotics of the spectral shift function ssf for schrodinger operator is studied at trapping energies it is shown that the ssf converges to sum of a smooth function and a step function which is essentially the counting function of resonances in particular the weyl asymptotics is proved

eigenvalues of the reference operator and semiclassical resonances we prove that the estimate of the number of the eigenvalues in intervals 0 0 and the number of the positive eigenvalues of l h in we apply this result to obtain a breit wigner approximation of the derivative of the spectral shift function near critical energy levels

integral homology of real flag manifolds and loop spaces of symmetrical spaces abstract an algorithm is given for the computation of the integral homology groups of generalized real flag manifolds and of based loop spaces of compact symmetric spaces this extends earlier work of bott and samelson computing these invariants with z 2 coefficients the algorithm uses the root system of the underlying symmetric spaces to turn the computation into a relatively straightforward combinatorial calculation the proofs of the theorems use morse theory of the moment map as tool to understand the geometry of the spaces in question

semiclassical asymptotics gauge fields and quantum chaos a gauge field is given by a connection on a principal bundle p m we consider the semiclassical behavior of a family of schrodinger operators associated with a gauge field in the limit as h 0 we relate the spectral theory of such operators to behavior of the hamiltonian flow on the natural phase space associated to a gauge field examining in particular situations where this flow exhibits chaotic behavior

breit wigner formula at barrier tops for noncritical energies the asymptotic behavior of the scattering phase and of the time delay are known to be described by a weyl type formula and the breit wigner formula respectively we consider here the case of critical energy levels in dimension 1 we obtain the semiclassical asymptotics of the scattering phase and of the time delay uniformly with respect to the energy in a neighborhood of a critical value

ergodicite et limite semi classique consider a self adjoint quantic hamiltonian p h p x hdx whereh 0 is the planck s constant andp some smooth classical observable on the phase space r2n when the classical flow on a compact energy shell p is ergodic we prove that in the limith 0 almost all the eigenfunctions ofp h whose energy is near of are distributed according to the liouville measure on p

stationary patterns of a prey predator system with a protection zone and cross diffusion of fractional type abstract this paper is concerned with the stationary patterns of a prey predator model with a protection zone and fractional type cross diffusion for the prey it is shown that the fractional type cross diffusion has negative effects on the survival of the prey when the intrinsic growth rate of the predator is positive moreover our mathematical analysis shows that compared with the results obtained in k oeda 2011 and k oeda 2012 the large cross diffusion coefficient and large growth rate of the predator species have some essentially different effects on the profiles of the solutions

intermediate schauder theory for second order parabolic equations i estimates on introduit les normes et les espaces de fonctions a utiliser les estimations a priori pour les problemes a coefficients constants dans des domaines simples sont reduits a des estimations integrales on donne des conditions sous lesquelles on peut passer localement a des coefficients variables on donne des estimations globales

semiclassical resolvent estimates for trapping perturbations we study the semiclassical estimates of the resolvent of a self adjoint operator l h in the space of bounded operators in the general case of long range trapping black box perturbations we prove that the estimate of the cut off resolvent implies the estimate

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

approximation semi classique de l equation de heisenberg we study the semi classical approximation for the solution of heisenberg equation in terms of pseudo differential operators and establish a semi classical version of egorov s theorem as an application of these results we get the classical limit of quantum mechanical correlation functions for a class of non bounded observables

spatial pattern in an ecosystem of phytoplankton nutrient from remote sensing abstract in this paper we consider an ecosystem of phytoplankton nutrient which comes from pelagic zone of the ocean by using remote sensing we are concerned with the pattern formation of the model we first give the asymptotic behavior of positive solutions of parabolic dynamical system then we discuss the stability of constant positive solution and the boundedness of stationary solutions finally the non existence and existence results of the pattern are investigated by considering some parameters the result shows that under certain range of these parameters the pattern of the model occurs

on the indices of fixed points of mappings in cones and applications abstract assume that k is a cone in a banach space and a k k is completely continuous we obtain a formula for the index in k of a fixed point of a under the assumption that a linearization exists and satisfies an invertibility condition we then use this formula to generalize some results of amann on the number of fixed points of a to the case where k has empty interior

the spectrum of the scattering matrix near resonant energies in the semiclassical limit the object of study in this paper is the on shell scattering matrix s e of the schrodinger operator with the potential satisfying assumptions typical in the theory of shape resonances we study the spectrum of s e in the semiclassical limit when the energy parameter e varies from eres e to eres e where eres is a real part of a resonance and e is sufficiently small the main result of our work describes the spectral flow of the scattering matrix through a given point on the unit circle this result is closely related to the breit wigner effect

breit wigner formulas for the scattering phase and the total scattering cross section in the semi classical limit in this paper we prove results in resonance scattering for the schrodinger operatorpv h2 v v being a smooth short range potential onrn more precisely for energy near a trapping energy level 0 for the classical system defined by the hamiltonianp x 2 v x we prove that the scattering phase and the scattering cross sections associated to pv p0 have the breit wigner form lorentzian line shape in the limith 0

spectral distributions for long range perturbations abstract we study distributions which generalize the concept of spectral shift function for pseudo differential operators on r d we call such distributions spectral distributions relations between relative scattering determinants and spectral distributions are established they lead to the definition of regularized scattering phase these relations are analogous to the usual one for the standard spectral shift function we give several asymptotic properties in the high energy and semiclassical limits where both nontrapping and trapping cases are considered in particular we prove breit wigner formulae for the regularized scattering phases for semiclassical schrodinger operators with long range potentials

the shape resonance for a class of schrodinger operatorsh 2 2m v onl 2 n with potentials having minima embedded in the continuum of the spectrum and non trapping tails we show the existence of shape resonances exponentially close to the real axis as 0 the resonant energies are given by a convergent perturbation expansion in powers of a parameter exhibiting the expected exponentially small behaviour for tunneling

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

on nonlinear reaction diffusion systems abstract this paper presents a qualitative analysis for a coupled system of two reaction diffusion equations under various boundary conditions which arises from a number of physical problems the nonlinear reaction functions are classified into three basic types according to their relative quasi monotone property for each type of reaction functions an existence comparison theorem in terms of upper and lower solutions is established for the time dependent system as well as some boundary value problems three concrete physical systems arising from epidemics biochemistry and engineering are taken as representatives of the basic types of reacting problems through suitable construction of upper and lower solutions various qualitative properties of the solution for each system are obtained these include the existence and bounds of time dependent solutions asymptotic behavior of the solution stability and instability of nontrivial steady state solutions estimates of stability regions and finally the blowing up property of the solution special attention is given to the homogeneous neumann boundary condition

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

semi classical asymptotics for local spectral densities and time delay problems in scattering processes abstract we study the semi classical asymptotics for local spectral densities of schrodinger operators 1 2 h 2 v and apply the obtained result to the time delay problem in potential scattering theory it is shown that for a class of central potentials the time delay in quantum mechanics converges to the corresponding one in classical mechanics in the semi classical limit in a non trapping energy region however such a convergence is not expected in a trapping energy region

calcul fonctionnel sur les operateurs admissibles et application abstract let h v h h 2 2 v lim x v x under suitable conditions we prove that h v h is a pseudodifferential operator whose symbol has an asymptotic a h j 0 h j a j more general pseudodifferential operator s classes are also considered we apply this result to study the semi classical behaviour of the spectrum of h v as h 0 so we improve recent results obtained by j chazarain and by the author in collaboration with b helffer furthermore we give a precise meaning to the formal development considered in b grammaticos and a voros work ann physics 123 1979 359 380

time decay of scattering solutions and resolvent estimates for semiclassical schrodinger operators we study the semiclassical schrodinger operator h2 v x h 0 1 and establish various results on the time decay of scattering solutions and the semiclassical bounds for the powers of the resolvent for short range potentials the time decay results obtained in this paper are the best possible and this enables us to conclude the equivalence between the non trapping condition in classical mechanics and the uniform time decay of wave functions in quantum mechanics

feynman path integrals and the trace formula for the schrodinger operators we study schrodinger operators of the form h frac h 2 2 delta frac 1 2 x cdot a 2 x v x on d wherea2 is a strictly positive symmetricd d matrix andv x is a continuous real function which is the fourier transform of a bounded measure if n are the eigenvalues ofh we show that the theta function theta t sum limits n exp left frac i h t lambda n right is explicitly expressible in terms of infinite dimensional oscillatory integrals feynman path integrals over the hilbert space of closed trajectories we use these explicit expressions to give the asymptotic behaviour of t for smallh in terms of classical periodic orbits thus obtaining a trace formula for the schrodinger operators this then yields an asymptotic expansion of the spectrum ofh in terms of the periodic orbits of the corresponding classical mechanical system these results extend to the physical case the recent work on poisson and trace formulae for compact manifolds

semiclassical asymptotics for the spectral function of long range schrodinger operators we obtain semiclassical asymptotics for the spectral function of schrodinger operators with long range potentials we study the case of non trapping energy levels and also the case of a potential well inside an island where resonances due to trapped trajectories appear

stationary patterns for a prey predator model with prey dependent and ratio dependent functional responses and diffusion abstract we study a reaction diffusion system arising from a three species prey predator model with prey dependent and ratio dependent functional responses the main aim of this paper is to study the global existence and bifurcation of non constant positive steady states in particular we will show that even though the unique positive constant steady state is stable for the ode dynamics non constant positive steady states exist for the reaction diffusion system this demonstrates that stationary patterns arise as a result of diffusion

space time picture of semiclassical resonances this paper addresses the theory of quasiclassical resonances for schrodinger operators with potentials smooth outside some possible local singularities we introduce the notion of quasiresonance similar to that of quasimode but incorporating a condition revealing its scattering nature and describe its space time behaviour the definition is given in terms of the original schrodinger operator and uses a description of its frequency set the result on the space time behaviour justifies the intuitive picture of resonances as metastable states or bound states with finite life times we demonstrate how quasiresonances arise in several natural situations

coexistence theorems of steady states for predator prey interacting systems on donne des conditions necessaires et suffisantes pour l existence des solutions positives des etats stationnaires pour des systemes predateur proie sous des conditions aux limites de dirichlet sur r n

existence and instability of neumann layer solutions for a 3 component lotka volterra model with diffusion abstract we consider a 3 component lotka volterra model with diffusion which describes the dynamics of the population of two competing prey and one predator and we discuss the existence of positive stationary solutions and their stability property to do this the singular perturbation method and the associated singular limit spectral analysis are employed

on the geometry of semiclassical limits on dirichlet spaces abstract this paper is a contribution to semiclassical analysis for abstract schrodinger type operators on locally compact spaces let x be a metrizable separable locally compact space let be a radon measure on x with a full support let t x y p t x y be a strictly positive pointwise consistent heat kernel and assume that the generator h p 0 of the corresponding self adjoint contraction semigroup in l 2 x induces a regular dirichlet form then given a function 0 1 0 such that the limit lim t 0 p t x x t exists for all x x we prove that for every potential w x r one has lim t 0 t tr e t h p w e w x lim t 0 p t x x t d x for the schrodinger type operator h p w provided w satisfies very mild conditions at that are essentially only made to guarantee that the sum of quadratic forms h p w t is self adjoint and bounded from below for small t and to guarantee that e w x lim t 0 p t x x t d x the proof is probabilistic and relies on a principle of not feeling the boundary for p t x x in particular this result implies a new semiclassical limit result for partition functions valid on arbitrary connected geodesically complete riemannian manifolds and one also recovers a previously established semiclassical limit result for possibly locally infinite connected weighted graphs