phase space tunneling for operators with symbols in a gevrey class phase space tunneling and exponential decay of eigenfunctions in phase space are well known for operators with symbols which are analytic in some neighborhood of the real axis this can be used to prove an adiabatic theorem of exponential order if one assumes the hamiltonian to depend analytically on time however to study compactly supported switching processes one has to weaken the analyticity assumptions here we examine nonanalytic symbols with gevrey class regularity and show that we get an exponential decay of the corresponding eigenfunctions with respect to 1 a as 0 where a 1 the loss of regularity causes a slower decay in the analysis is done using the methods of martinez and its generalization by nakamura an upper bound for the rate of decay is given

ramification of local fields with imperfect residue fields we define two decreasing filtrations by ramification groups on the absolute galois group of a complete discrete valuation field whose residue field may not be perfect in the classical case where the residue field is perfect we recover the classical upper numbering filtration the definition uses rigid geometry and log structures we also establish some of their properties

some properties of a random walk on a comb structure abstract we analyze transport properties of a random walk on a comb structure which serves as a model for a random walk on the backbone of a percolation cluster it is shown that the random walk along the x axis which is the analog of the backbone exhibits anomalous diffusion in that x 2 n n 1 2 and the expected number of x sites visited is proportional to n 1 4 for large n the distribution function is found to be a two dimensional gaussian if a field in the x direction so that diffusion is asymmetric the expected displacement is found to be asymptotically proportional to n 1 2

estimations de la resolvante pour une molecule diatomique dans l approximation de born oppenheimer making use of an adiabatic operator that takes several electronic states into account we derive a born oppenheimer approximation of the resolvent for a diatomic molecule this is an improvement of a result in kmw1 such a resolvent approximation is useful to obtain an adiabatic approximation of total cross sections see jec2 the strategy we use based on mourre s commutator method and on a new kind of global escape function may be carried over to control the resolvent of some matricial schrodinger operators in the same way we obtain a semiclassical estimate for the resolvent of the semiclassical dirac operator with scalar electric potential extending a result of ce

resonances of polyatomic molecules in the born oppenheimer approximation we study the spectral properties of polyatomic molecules near a scattering level we prove that in the born oppenheimer approximation this study can be reduced to the one of a family of finite matrices of semiclassical pseudodifferential operators more precisely we show that any resonance of the molecule which is close enough to the real axis can be obtained from the discrete spectrum of one of these matrixes

magnetic wkb constructions this paper is devoted to the semiclassical magnetic laplacian until now wkb expansions for the eigenfunctions were only established in the presence of a non zero electric potential here we tackle the pure magnetic case thanks to feynman hellmann type formulas and coherent states decomposition we develop here a magnetic born oppenheimer theory exploiting the multiple scales of the problem we are led to solve an effective eikonal equation in pure magnetic cases and to obtain wkb expansions we also investigate explicit examples for which we can improve our general theorem global wkb expansions quasi optimal agmon estimates and upper bound of the tunelling effect in symmetric cases we also apply our strategy to get more accurate descriptions of the eigenvalues and eigenfunctions in a wide range of situations analyzed in the last two decades

1 2 order fractional fokker planck equation on comblike model from the generalized scheme of random walks on the comblike structure it is shown how a 1 2 order fractional fokker planck equation can be derived the operator method for the moments associated with the distribution function p x t is used to solve the resulting equation also the anomalous diffusion along the backbone of the structure has been considered

a globally convergent newton gmres subspace method for systems of nonlinear equations newton krylov methods are variants of inexact newton methods where the approximate newton direction is taken from a subspace of small dimension here we introduce a new hybrid newton gmres method where a global strategy restricted to a low dimensional subspace generated by gmres is performed the obtained process is consistent with preconditioning and with matrix free implementation computational results indicate that our proposal enhances the classical backtracking inexact method

a constrained optimization approach to solving certain systems of convex equations abstract this research presents a new constrained optimization approach for solving systems of nonlinear equations particular advantages are realized when all of the equations are convex for example a global algorithm for finding the zero of a convex real valued function of one variable is developed if the algorithm terminates finitely then either the algorithm has computed a zero or determined that none exists if an infinite sequence is generated either that sequence converges to a zero or again no zero exists for solving n dimensional convex equations the constrained optimization algorithm has the capability of determining that the system of equations has no solution global convergence of the algorithm is established under weaker conditions than previously known and in this case the algorithm reduces to newton s method together with a constrained line search at each iteration it is also shown how this approach has led to a new algorithm for solving the linear complementarity problem

a finite improvement algorithm for the linear complementarity problem abstract this paper is concerned with the development of an improvement algorithm for the linear complementarity problem lcp our approach to find a solution to lcp is to solve the equivalent constrained optimization problem cop of maximizing the sum of the minimum of each complementary pair of variables subject to the constraints that each such minimum is nonpositive the algorithm ascent in nature is similar to the simplex method in the sense that it moves between basic points of an associated system of linear equations these basic points are feasible to our cop whose objective function is piecewise linear and concave classes of matrices are characterized for which our algorithm processes lcp for every right hand side vector and every matrix in the class a computational study shows that our algorithm is clearly superior to a previos improvement algorithm computational comparisons with lemke s well known algorithm are also presented

diffusion on a random comb distribution function of the survival probability we investigate the distribution functionq p describing the survival probability on a comb consisting of a backbone with lateral randomly disconnected infinite branches two different regimes are analyzed in some detail i at short times q p is shown to have a self similar structure devil s staircase ii at large times this function becomes smooth and tends toward a rather well defined unit step function the disorder averaged survival probability is expected to decrease ast 3 4 at large times whereas the relative fluctuations of the sample dependentp0 t display a very slow decay in time going to zero liket 1 8

exponentially accurate semiclassical asymptotics of low lying eigenvalues for 2 2 matrix schrodinger operators abstract we consider a simple molecular type quantum system in which the nuclei have one degree of freedom and the electrons have two levels the hamiltonian has the form h 4 2 2 y 2 h y where h y is a 2 2 real symmetric matrix near a local minimum of an electron level e y that is not at a level crossing we construct quasimodes that are exponentially accurate in the square of the born oppenheimer parameter by optimal truncation of the rayleigh schrodinger series that is we construct e and such that o 1 and h e exp 2 where 0

analytic study of a model of diffusion on a random comblike structure abstract we present an analytic study of a model of diffusion on a random comblike structure in which no bias field does exist neither along the backbone nor along the branches for any given disordered structure our analytic treatment allows to compute in an exact manner the asymptotic behaviour at large time of the probability of presence of the particle at its initial site and on the backbone and of the particle position and dispersion the expressions of these quantities are shown to coincide asymptotically with those derived in simple mean field treatments the results for any given sample do not depend on the particular configuration self averaging property the behaviours strongly depend on the distribution of the lengths of the branches with an exponential distribution transport is normal while anomalous diffusion may take place for a power law distribution when long branches are present with a sufficiently high weight

on the born oppenheimer expansion for polyatomic molecules we consider the schrodinger operatorp h for a polyatomic molecule in the semiclassical limit where the mass ratioh 2 of electronic to nuclear mass tends to zero we obtain wkb type expansions of eigenvalues and eigenfunctions ofp h to all orders inh this allows to treat the splitting of the ground state energy of a non planar molecule our class of potentials covers the physical case of the coulomb interaction we use methods ofh pseudodifferential operators with operator valued symbols which by use of appropriate coordinate changes in local coordinate patches covering the classically accessible region become applicable even to our class of singular potentials

the infinite time limit for the time dependent born oppenheimer approximation we analyze the schrodinger equation i varepsilon 2 tfrac partial partial t psi h varepsilon psi whereh is the hamiltonian of the molecular system consisting of nuclei with masses proportional to 4 and electrons with masses of order 1 using the born oppenheimer method we construct the leading order asymptotic expansion to the exact solutions of the equation we show that if the particles interact through smooth potentials decaying suitably as the distance between particles tends to then the expansion holds uniformly for all timest 0 by similar analysis one can show validity of the expansion fort 0 thus our results hold for scattering theory

adiabatic decoupling and time dependent born oppenheimer theory we reconsider the time dependent born oppenheimer theory with the goal to carefully separate between the adiabatic decoupling of a given group of energy bands from their orthogonal subspace and the semiclassics within the energy bands band crossings are allowed and our results are local in the sense that they hold up to the first time when a band crossing is encountered the adiabatic decoupling leads to an effective schrodinger equation for the nuclei including contributions from the berry connection

semiclassical limit for multistate klein gordon systems almost invariant subspaces and scattering theory by using the method of helffer and sjostrand to construct moyal projections we extend the almost invariant subspace theory to the semiclassical context applications to the semiclassical limit for two component klein gordon hamiltonian are given more precisely under the conditions that the potential is analytic and its eigenvalues never cross we prove that the scattering matrix is block diagonal up to exponentially small errors also we show how the existence of almost invariant subspaces leads to the existence of quasimodes with exponentially long lifetimes

precise exponential estimates in adiabatic theory general adiabatic evolutions associated to hamiltonians which admit a holomorphic extension with respect to the time variable in a complex strip and whose spectrum satisfies a gap condition are studied an explicit rate of exponential decay is given which is related to simple geometric quantities associated to the spectrum of the hamiltonian for the transition probability between the two parts of the spectrum when the evolution is taken from to

molecular predissociation resonances near an energy level crossing ii vector field interaction abstract we study the resonances of a two by two semiclassical system of one dimensional schrodinger operators near an energy where the two potentials intersect transversally one of them being bonding and the other one anti bonding assuming that the interaction is a vector field we obtain optimal estimates on the location and on the widths of these resonances

molecular dynamics at an energy level crossing abstract this paper is a continuation of a previous work 2 about the study of the survival probability modelizing the molecular predissociation in the born oppenheimer framework here we consider the critical case where the reference energy corresponds to the value of a crossing of two electronic levels one of these two levels being confining while the second dissociates we show that the survival probability associated to a certain initial state is a sum of the usual time dependent exponential contribution and a reminder term that is jointly polynomially small with respect to the time and the semiclassical parameter we also compute explicitly the main contribution of the remainder

resonances dans l approximation de born oppenheimer i resume on etudie l operateur p h2 x y v x y sur r xn r yp lorsque h tend vers zero dans un cas ou des resonances apparaissent en utilisant la methode de feshbach l etude de p est d abord ramenee a celle d un operateur matriciel sur l2 r xn de terme principal diag h2 j x ou les j x sont les valeurs propres de y v x y sur l2 r yp sous l hypothese que 2 admet un puits ponctuel non degenere et des conditions supplementaires sur 1 il est alors prouve que p possede des resonances ayant un developpement asymptotique reel en h 1 2 proches des valeurs propres de h2 2 x

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

microlocal wkb expansions in relation with microlocal tunneling we investigate the existence of micro local wkb expansions for the eigenfunctions of semiclassical pseudodifferential operators whose symbol admits a non degenerate minimum at some point x0 0 of r2n more precisely under assumptions of analyticity we prove that the fbi or bargman transform of the first eigenfunction of such an operator admits near x0 0 an expansion of the form e x h j 0 hjaj x as the semiclassical parameter h tends to 0 here and the aj s are smooth complex functions and the domain in which the expansion is valid can be described in terms of deformation properties our result can be applied e g to magnetic schrodinger operators with electric potential admitting a non degenerate well

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

newton s method for linear complementarity problems this paper presents an iterative newton type method for solving a class of linear complementarity problems this class was discovered by mangasarian who had established that these problems can be solved as linear programs cottle and pang characterized solutions of the problems in terms of least elements of certain polyhedral sets the algorithms developed in this paper are shown to converge to the least element solutions some applications and computational results are also discussed

phase space tunneling in multistate scattering abstract we prove exponentially small estimates on the off diagonal terms of the scattering matrix associated to two state semiclassical schrodinger hamiltonians our method is based on phase space tunneling estimates and a splitting of the operator by means of toeplitz type phase space cutoff operators