ground state of the massless nelson model in a non fock representation we consider a model of a particle coupled to a massless scalar field the massless nelson model in a non fock representation we prove the existence of a ground state of the system applying the method of griesemer lieb and loss

lowest energy states in nonrelativistic qed atoms and ions in motion within the framework of nonrelativisitic quantum electrodynamics we consider a single nucleus and n electrons coupled to the radiation field since the total momentum p is conserved the hamiltonian h admits a fiber decomposition with respect to p with fiber hamiltonian h p a stable atom resp ion means that the fiber hamiltonian h p has an eigenvalue at the bottom of its spectrum we establish the existence of a ground state for h p under i an explicit bound on p ii a binding condition and iii an energy inequality the binding condition is proven to hold for a heavy nucleus and the energy inequality for spinless electrons

exact multiplicity and stability of solutions of second order neumann boundary value problem abstract the purpose of this paper is to establish the exact multiplicity and stability of solutions of the equation u g x u f x with the neumann boundary value conditions u 0 u 1 0 exactly three ordered solutions are obtained by taking advantage of the anti maximum principle combined with the methods of upper and lower solutions moreover we obtain that one of three solutions is negative while the other two are positive the middle solution is unstable and the remaining two are stable

periodic solutions of a newtonian equation stability by the third approximation abstract this paper presents a sufficient condition for the stability of periodic solutions of a newtonian equation this condition depends on the third order approximation and does not involve small parameters an application to an equation with cubic potential is given

time ordered exponential on the complex plane and gell mann low formula as a mathematical theorem the time ordered exponential representation of a complex time evolution operator in the interaction picture is studied using the complex time evolution we prove the gell mann low formula under certain abstract conditions in mathematically rigorous manner we apply the abstract results to quantum electrodynamics with cutoffs

blow up phenomena of second order nonlinear differential equations abstract this paper studies the blow up phenomena of the second order nonlinear differential equation x f t x x under appropriate conditions it is shown that some solutions of the equation blow up in finite time

smoothness and analyticity of perturbation expansions in qed abstract we consider the ground state of an atom in the framework of non relativistic qed we assume that the ultraviolet cutoff is of the order of the rydberg energy and that the atomic hamiltonian has a non degenerate ground state we show that the ground state energy and the ground state are k times continuously differentiable functions of the fine structure constant and respectively the square root of the fine structure constant on some nonempty interval 0 c k

relativistic schrodinger operators asymptotic behavior of the eigenfunctions nonrelativistic schrodinger operators are perturbations of the negative laplacian and the connection with stochastic processes and brownian motion in particular is well known and usually goes under the name of feynman and kac we present a similar connection between a class of relativistic schrodinger operators and a class of processes with stationary independent increments in particular we investigate the decay of the eigenfunctions of these operators and we show that not only exponential decay but also polynomial decay can occur

plane polynomial automorphisms of fixed multidegree let mathcal g be the group of polynomial automorphisms of the complex affine plane on one hand mathcal g can be endowed with the structure of an infinite dimensional algebraic group see shafarevich in math ussr izv 18 214 226 1982 and on the other hand there is a partition of mathcal g according to the multidegree see friedland and milnor in ergod th dyn syst 9 67 99 1989 let mathcal g d denote the set of automorphisms whose multidegree is equal to d we prove that mathcal g d is a smooth locally closed subset of mathcal g and show some related results we give some applications to the study of the varieties mathcal g m resp mathcal g leq m of automorphisms whose degree is equal to m resp is less than or equal to m

singularly perturbed higher order periodic boundary value problems abstract the existence of positive periodic solutions is proved for the higher order singular differential equation 0 1 x n a n 1 x n 1 a 1 x a 0 t x e t and for some generalizations including a class of differential systems with singular cyclic feedback

bifurcations of subharmonics abstract we consider a piecewise linear second order ordinary differential equation which is topologically equivalent to the sine gordon equation the system is subjected to a time harmonic disturbance and the behavior of the periodic solutions is examined it is shown that subharmonic motions with period n times that of the disturbance appear via saddle node bifurcations for all n 1 2 3 these motions then undergo period doubling or pitchfork bifurcations as parameters are varied beyond the saddle node bifurcation values the limit n is considered and is compared with a melnikov calculation for the system due to the piecewise linear nature of the system no approximations are necessary and thus the results are valid for nonsmall parameter values furthermore this provides an answer to the uniformity question in the study of stability of subharmonics

effective density of states for a quantum oscillator coupled to a photon field we give an explicit formula for the effective partition function of a harmonically bound particle minimally coupled to a photon field in the dipole approximation the effective partition function is shown to be the laplace transform of a positive borel measure the effective measure of states the absolutely continuous part of the latter allows for an analytic continuation the singularities of which give rise to resonances we give the precise location of these singularities and show that they are well approximated by first order poles with residues equal to the multiplicities of the corresponding eigenspaces of the uncoupled quantum oscillator thus we obtain a complete analytic description of the natural line spectrum of the charged oscillator

exact multiplicity for periodic solutions of duffing type abstract in this paper we study the following duffing type equation x cx g t x h t where g t x is a 2 periodic continuous function in t and concave convex in x and h t is a small continuous 2 periodic function the exact multiplicity and stability of periodic solutions are obtained

gibbs measures relative to brownian motion we consider brownian motion perturbed by the exponential of an action the action is the sum of an external one body potential and a two body interaction potential which depends only on the increments under suitable conditions on these potentials we establish existence and uniqueness of the corresponding gibbs measure we also provide an example where uniqueness fails because of a slow decay in the interaction potential

rate of decay of stable periodic solutions of duffing equations abstract in this paper we consider the second order equations of duffing type bounds for the derivative of the restoring force are given that ensure the existence and uniqueness of a periodic solution furthermore the stability of the unique periodic solution is analyzed the sharp rate of exponential decay is determined for a solution that is near to the unique periodic solution

topological degree in the generalized gause prey predator model abstract we consider a generalized gause prey predator model with t periodic continuous coefficients in the case where the poincare map p over time t is well defined the result of the paper can be explained as follows we locate a subset u of r 2 such that the topological degree d i p u equals to 1 the novelty of the paper is that the later is done under only continuity and some monotonicity assumptions for the coefficients of the model a suitable integral operator is used in place of the poincare map to cope with possible nonuniqueness of solutions the paper therefore provides a new framework for studying the generalized gause model with functional differential perturbations and multi valued ingredients

on the pumping effect in a pipe tank flow configuration with friction abstract we provide sufficient conditions for the existence and the asymptotic stability of periodic positive solutions for a pipe tank flow configuration the model is a nonlinear ordinary second order differential equation with a singularity containing the second power of the derivative of the unknown function and with a linear term representing friction in this way we complement previous results obtained by g propst in 2006

functional integral approach to semi relativistic pauli fierz models abstract by means of functional integrations spectral properties of semi relativistic pauli fierz hamiltonians h p a 2 m 2 m v h rad in quantum electrodynamics are considered here p is the momentum operator a a quantized radiation field on which an ultraviolet cutoff is imposed v an external potential h rad the free field hamiltonian and m 0 describes the mass of electron two self adjoint extensions of a semi relativistic pauli fierz hamiltonian are defined the feynman kac type formula of e t h is given a self adjointness a spatial decay of bound states a gaussian domination of the ground state and the existence of a measure associated with the ground state are shown all the results are independent of values of coupling constant and it is emphasized that m 0 is included

stability and bifurcations for dissipative polynomial automorphisms of mathbb c 2 we study stability and bifurcations in holomorphic families of polynomial automorphisms of mathbb c 2 we say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there we first show that this defines a meaningful notion of stability which parallels in many ways the classical notion of j stability in one dimensional dynamics define the bifurcation locus to be the complement of the weak stability locus in the second part of the paper we prove that under an assumption of moderate dissipativity the parameters displaying homoclinic tangencies are dense in the bifurcation locus this confirms one of palis conjectures in the complex setting the proof relies on the formalism of semi parabolic bifurcation and the construction of critical points in semi parabolic basins which makes use of the classical denjoy carleman ahlfors and wiman theorems

exponential decay of eigenfunctions of brown ravenhall operators we prove the exponential decay of eigenfunctions of reductions of brown ravenhall operators to arbitrary irreducible representations of rotation reflection and permutation symmetry groups under the assumption that the corresponding eigenvalues are below the essential spectrum

resolvent smoothness and local decay at low energies for the standard model of non relativistic qed we consider an atom interacting with the quantized electromagnetic field in the standard model of non relativistic qed the nucleus is supposed to be fixed we prove smoothness of the resolvent and local decay of the photon dynamics for quantum states in a spectral interval i just above the ground state energy our results are uniform with respect to i their proofs are based on abstract mourre s theory a mourre inequality established by frohlich griesemer and sigal see frohlich et al 2008 14 hardy type estimates in fock space and a low energy dyadic decomposition

infrared renormalization in non relativistic qed and scaling criticality abstract we consider a spin 1 2 electron in the framework of non relativistic quantum electrodynamics qed let h p denote the fiber hamiltonian corresponding to the conserved total momentum p r 3 of the electron and the photon field regularized by a fixed ultraviolet cutoff in the interaction term and an infrared regularization parametrized by 0 1 ultimately our goal is to remove the latter by taking 0 we prove that there exists a constant 0 0 1 independent of 0 such that for all p 1 3 and all values of the finestructure constant 0 0 there exists a ground state eigenvalue of multiplicity two at the bottom of the essential spectrum moreover we prove that the renormalized electron mass satisfies 1 m ren p 1 c uniformly in 0 in units where the bare mass has the value 1 and we prove the existence of the renormalized mass in the limit 0 our analysis uses the isospectral renormalization group method of bach frohlich sigal introduced in v bach j frohlich i m sigal quantum electrodynamics of confined non relativistic particles adv math 137 2 1998 299 395 v bach j frohlich i m sigal renormalization group analysis of spectral problems in quantum field theory adv math 137 1998 205 298 and further developed in v bach t chen j frohlich i m sigal smooth feshbach map and operator theoretic renormalization group methods j funct anal 203 1 2003 44 92 v bach t chen j frohlich i m sigal the renormalized electron mass in non relativistic qed j funct anal 243 2 2007 426 535 the limit 0 determines a scaling critical or endpoint type renormalization group problem in which the interaction is strictly marginal of scale independent size a main result of this paper is the development of a method that provides rigorous control of the renormalization of a strictly marginal quantum field theory characterized by a non trivial scaling limit the key ingredients entering this analysis include a hierarchy of exact algebraic cancelation identities exploiting the spatial and gauge symmetries of the model and a combination of the isospectral renormalization group method with the strong induction principle

on the number of solutions for the forced pendulum equation soit e e t c 0 t une fonction t periodique avec 0 t e s ds 0 on considere l equation x t x t ksin x t e t c x est t periodique ou k c r et k 0

spectrum of the semi relativistic pauli fierz model i abstract a hvz type theorem for the semi relativistic pauli fierz hamiltonian h p 1 a 2 m 2 1 v 1 1 h f m 0 in quantum electrodynamics is studied here h is a self adjoint operator in hilbert space l 2 r d f r d f d x a r d a x d x is a quantized radiation field and h f is the free field hamiltonian defined by the second quantization of a dispersion relation r d r it is emphasized that massless case m 0 is included let e inf h be the bottom of the spectrum of h suppose that the infimum of is m 0 then it is shown that ess h e m in particular the existence of the ground state of h can be proven

ground states and spectrum of quantum electrodynamics of nonrelativistic particles a system consisting of finitely many nonrelativistic particles bound on an external potential and minimally coupled to a massless quantized radiation field without the dipole approximation is considered an ultraviolet cut off is imposed on the quantized radiation field the hamiltonian of the system is defined as a self adjoint operator in a hilbert space the existence of the ground states of the hamiltonian is established it is shown that there exist asymptotic annihilation and creation operators hence the location of the absolutely continuous spectrum of the hamiltonian is specified

on enhanced binding and related effectsin the non and semi relativistic pauli fierz models we prove enhanced binding and increase of binding energies in the non and semi relativistic pauli fierz models for arbitrary values of the fine structure constant and the ultra violet cut off and discuss the resulting improvement of exponential localization of ground state eigenvectors for the semi relativistic model we also discuss the increase of the renormalized electron mass and determine the linear leading order term in the asymptotics of the self energy as the ultra violet cut off goes to infinity

global dynamics of delay equations for populations with competition among immature individuals abstract we analyze a population model for two age structured species allowing for inter and intra specific competition at immature life stages the dynamics is governed by a system of delay differential equations ddes recently introduced by gourley and liu the analysis of this model presents serious difficulties because the right hand sides of the ddes depend on the solutions of a system of nonlinear odes and generally cannot be solved explicitly using the notion of strong attractor we reduce the study of the attracting properties of the equilibria of the ddes to the analysis of a related two dimensional discrete system then we combine some tools for monotone planar maps and planar competing lotka volterra systems to describe the dynamics of the model with three different birth rate functions we give easily verifiable conditions for global extinction of one or the two species and for global convergence of the positive solutions to a coexistence state

functional integral representations of the pauli fierz model with spin 1 2 a feynman kac type formula for a levy and an infinite dimensional gaussian random process associated with a quantized radiation field is derived in particular a functional integral representation of e thpf generated by the pauli fierz hamiltonian with spin 1 2 in non relativistic quantum electrodynamics is constructed when no external potential is applied hpf turns translation invariant and it is decomposed as a direct integral hpf r3 hpf p dp the functional integral representation of e thpf p is also given although all these hamiltonians include spin nevertheless the kernels obtained for the path measures are scalar rather than matrix expressions as an application of the functional integral representations energy comparison inequalities are derived

spin boson model through a poisson driven stochastic process we give a functional integral representation of the semigroup generated by the spin boson hamiltonian by making use of a poisson point process and a euclidean field we present a method of constructing gibbs path measures indexed by the full real line which can be applied also to more general stochastic processes with jump discontinuities using these tools we then show existence and uniqueness of the ground state of the spin boson and analyze ground state properties in particular we prove super exponential decay of the number of bosons gaussian decay of the field operators derive expressions for the positive integer fractional and exponential moments of the field operator and discuss the field fluctuations in the ground state

three dimensional periodic systems with trivial dynamics ii 3 where t g r u s u u u g v v r is open and f t u 1 2 3 f t u u f t u f t u u is continuous in v is t periodic in the i 1 2 2 3 2 3 time t and has continuous partial derivative fr u defined on r v we also assume that the following hypotheses hold h f r u f r u f r u and f r u 0 on r v 1 1 2 2 1 2 3 3 2 3 h f has a continuous extension f r v a r such that 2 0 0 f r u is defined and continuous in a neighborhood v of v 0 0

fiber hamiltonians in non relativistic quantum electrodynamics abstract a translation invariant hamiltonian h of non relativistic quantum electrodynamics is studied this hamiltonian is decomposed with respect to the total momentum p t h r d h p d p where the self adjoint fiber hamiltonian h p is defined for arbitrary values of coupling constants the relationship between rotation invariance of h p and polarization vectors is discussed and functional integral representations of n point euclidean green functions of h p are given from these applications into the degeneracy of ground states ground state energy and expectation values of suitable observables with respect to ground states are given

quantum electrodynamics of confined nonrelativistic particles abstract we consider a system of finitely many nonrelativistic quantum mechanical electrons bound to static nuclei the electrons are minimally coupled to the quantized electromagnetic field but we impose an ultraviolet cutoff on the electromagnetic vector potential appearing in covariant derivatives and the interactions between the radiation field and electrons localized very far from the nuclei are turned off for a class of hamiltonians we prove exponential localization of bound states establish the existence of a ground state and derive sufficient conditions for its uniqueness furthermore we show that excited bound states of the unperturbed system become unstable and turn into resonances when the electrons are coupled to the radiation field to this end we develop a novel renormalization transformation which acts directly on the space of hamiltonians

asymptotic stability of forced oscillations emanating from a limit cycle abstract classical conditions for asymptotic stability of periodic solutions bifurcating from a limit cycle rely on the sign of the derivative of the associated bifurcation function at a zero in this paper we show that for analytic systems this result is of topological nature this means that it is enough to impose a change of sign at the zero without any assumption on the succesive derivatives

the number of periodic solutions of 2 dimensional periodic systems on donne des conditions suffisantes sur un systeme a 2 dimensions periodique qui a un nombre fini de solutions periodiques

stability and instability of standing waves to a system of schrodinger equations with combined power type nonlinearities abstract we obtain the existence of standing wave solutions to a coupled nonlinear schrodinger system i u 1 t u 1 u 1 p k 1 m k u 1 p k 2 u 2 q k u 1 in r n 0 t i u 2 t u 2 u 2 p k 1 m k u 1 p k u 2 q k 2 u 2 in r n 0 t u 1 x 0 u 10 x u 2 x 0 u 20 x in r n where 0 p 4 n 2 and k k p k q k are real constants with 1 p k p 1 1 q k p 1 p k q k p 2 and k q k k p k k 1 m we also discuss the stability and instability of the standing wave solutions

stability properties of periodic solutions of a duffing equation in the presence of lower and upper solutions we prove that a periodic solution of the duffing equationx cx g t x h t is asymptotically stable if and only if it is bracketed by a lower solution a and an upper solution b satisfying a t b t for every t provided that the derivative of g with respect to x is not too large we also produce a characterization of the asymptotic stability of the periodic solutions of the above equation in terms of certain stability properties of the corresponding fixed points of a related infinite dimensional order preserving discrete time dynamical system these results have a local flavour and therefore they naturally apply to the study of the stability in cases where g is not defined everywhere as a function of x or several periodic solutions exist as an application we briefly discuss the existence of stable and unstable periodic solutions for some classes of duffing equations with singular or oscillating nonlinearities

on the existence and uniqueness of ground states of a generalized spin boson model a generalization of the standard spin boson model is considered the hamiltonianh of the model with a coupling parameter racts in the tensor product h fbof a hilbert space h and the boson symmetric fock space fboverl2 r the existence and uniqueness of ground states ofh are investigated the degeneracy of the ground states is also discussed the results obtained arenonperturbative the methods used are those of constructive quantum field theory and the min max principle an exact asymptotic formula for the ground state energy ofh as is also established

ground states in non relativistic quantum electrodynamics the excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay but such a particle should have a true ground state one that minimizes the energy and satisfies the schrodinger equation we prove quite generally that this state exists for all values of the fine structure constant and the ultraviolet cutoff we also show the same thing for a many particle system under physically natural conditions

blow up solutions of cubic differential systems abstract we investigate the existence problem for blow up solutions of cubic differential systems we find sets of initial values of the blow up solutions we also discuss a method of finding upper estimates for the blow up time of these solutions our approach can be applied to systems of partial differential equations we apply this approach to the cauchy dirichlet problem for systems of semilinear heat equations with cubic nonlinearities

structure of the set of bounded solutions for a class of nonautonomous second order differential equations abstract some precise descriptions of the structure and behavior of bounded solutions for a class of nonautonomous second order differential equations are obtained the nonautonomous differential equations here are in the form x f t h x x where the driving system is defined in a compact base space h and the vector field f h x p is strictly increasing in state variable x on some interval a b the stability of the structure of bounded solutions with respect to parameter perturbations in f is also addressed

convergent expansions in non relativistic qed analyticity of the ground state we consider the ground state of an atom in the framework of non relativistic qed we show that the ground state as well as the ground state energy are analytic functions of the coupling constant which couples to the vector potential under the assumption that the atomic hamiltonian has a non degenerate ground state moreover we show that the corresponding expansion coefficients are precisely the coefficients of the associated raleigh schrodinger series as a corollary we obtain that in a scaling limit where the ultraviolet cutoff is of the order of the rydberg energy the ground state and the ground state energy have convergent power series expansions in the fine structure constant with dependent coefficients which are finite for 0

maximal velocity of photons in non relativistic qed abstract we consider the problem of propagation of photons in the quantum theory of non relativistic matter coupled to electromagnetic radiation which is presently the only consistent quantum theory of matter and radiation assuming that the matter system is in a localized state i e for energies below the ionization threshold we show that the probability to find photons at time t at the distance greater than c t where c is the speed of light vanishes as t as an inverse power of t

complex henon maps and discrete groups abstract consider the standard family of complex henon maps h x y p x a y x where p is a quadratic polynomial and a is a complex parameter let u be the set of points that escape to infinity under forward iterations the analytic structure of the escaping set u is well understood from previous work of j hubbard and r oberste vorth as a quotient of c d c by a discrete group of automorphisms isomorphic to z 1 2 z on the other hand the boundary j of u is a complicated fractal object on which the henon map behaves chaotically we show how to extend the group action to s 1 c in order to represent the set j as a quotient of s 1 c by an equivalence relation we analyze this extension for henon maps that are perturbations of hyperbolic polynomials with connected julia set or polynomials with a parabolic fixed point

some applications of hausdorff dimension inequalities for ordinary differential equations upper bounds are obtained for the hausdorff dimension of compact invariant sets of ordinary differential equations which are periodic in the independent variable from these are derived sufficient conditions for dissipative analytic n dimensional periodic differential equations to have only a finite number of periodic solutions for autonomous equations the same conditions ensure that each bounded semi orbit converges to a critical point these results yield some information about the lorenz equation and the forced duffing equation

renormalization analysis for degenerate ground states abstract we consider a hamilton operator which describes a finite dimensional quantum mechanical system with degenerate eigenvalues coupled to a field of relativistic bosons we show that the ground state projection and the ground state energy are analytic functions of the coupling constant in a cone with apex at the origin provided a mild infrared assumption holds to show the result operator theoretic renormalization is used and extended to degenerate situations

semi parabolic bifurcations in complex dimension two parabolic bifurcations in one complex dimension demonstrate a wide variety of interesting dynamical phenomena in this paper we consider the bifurcations of a holomorphic diffeomorphism in two complex dimensions with a semi parabolic semiattracting fixed point

exponential decay and ionization thresholds in non relativistic quantum electrodynamics abstract spatial localization of the electrons of an atom or molecule is studied in models of non relativistic matter coupled to quantized radiation we give two definitions of the ionization threshold one in terms of spectral data of cluster hamiltonians and one in terms of minimal energies of non localized states we show that these two definitions agree and that the electrons described by a state with energy below the ionization threshold are localized in a small neighborhood of the nuclei with a probability that approaches 1 exponentially fast with increasing radius of the neighborhood the latter result is derived from a new general result on exponential decay tailor made for our problem but applicable to many non relativistic quantum systems outside quantum electrodynamics as well

stochastic differential equations for models of non relativistic matter interacting with quantized radiation fields we discuss hilbert space valued stochastic differential equations associated with the heat semi groups of the standard model of non relativistic quantum electrodynamics and of corresponding fiber hamiltonians for translation invariant systems in particular we prove the existence of a stochastic flow satisfying the strong markov property and the feller property to this end we employ an explicit solution ansatz in the matrix valued case i e if the electron spin is taken into account it is given by a series of operator valued time ordered integrals whose integrands are factorized into annihilation preservation creation and scalar parts the feynman kac formula implied by these results is new in the matrix valued case furthermore we discuss stochastic differential equations and feynman kac representations for an operator valued integral kernel of the semi group as a byproduct we obtain analogous results for nelson s model

on global asymptotic stability of systems of lienard type we give new necessary and sufficient conditions under which the zero solution of lienard type systems is globally asymptotically stable to this end we examine i whether all trajectories intersect the vertical isocline or not and ii whether all trajectories tend to the origin or not we also apply our results to a pseudolinear system

ground state for point particles interacting through a massless scalar bose field we consider a massless scalar bose field interacting with two particles one of them infinitely heavy neither an infrared nor an ultraviolet cutoff is imposed in case the charge of the particles is of the same sign and sufficiently small we prove the existence of a ground state

a structure theorem for semi parabolic henon maps abstract consider the parameter space p c 2 of complex henon maps h c a x y x 2 c a y a x a 0 which have a semi parabolic fixed point with one eigenvalue e 2 i p q in this paper we provide a complete topological description of the julia set and forward julia set within a family of semi parabolic henon maps i e those henon maps from the curve p which are perturbations of a quadratic polynomial p with a parabolic fixed point of multiplier we prove that there is an open disk of parameters in p for which the semi parabolic henon map has connected julia set j and is structurally stable on j and j the forward julia set j has a nice local description inside a certain bidisk it is a trivial fiber bundle over j p the julia set of the polynomial p with fibers biholomorphic to a disk the julia set j is homeomorphic to the quotient of a solenoid by an equivalence relation and j j the closure of the saddle periodic points

dynamics of a periodic differential equation with a singular nonlinearity of attractive type where c g 0 a 0 and p is a continuous t periodic function for some t 0 we are interested in the existence and stability of positive t periodic solutions of 1 the existence of t periodic solutions of this class of equations where the restoring force is a singular nonlinearity that becomes infinite in zero w x has been proved by lazer and solimini 2 for the case without friction w x c s 0 and by habets and sanchez 1 for the damped case however the stability properties of these solutions have been less studied it is well known that for the autonomous case p t p 0 this equation has a 0 unique saddle point in this paper we prove that the dynamics of the periodic equation 1 is similar to the autonomous case for our study is n w x fundamental a massera s convergence theorem in r given by smith in 6 the paper is divided in three sections in section 1 the main results are stated it is seen that 1 has a unique unstable periodic solution w

essential self adjointness of translation invariant quantum field models for arbitrary coupling constants the hamiltonian of a system of quantum particles minimally coupled to a quantum field is considered for arbitrary coupling constants the hamiltonian has a translation invariant part by means of functional integral representations the existence of an invariant domain under the action of the heat semigroup generated by a self adjoint extension of the translation invariant part is shown with a non perturbative approach it is proved that the hamiltonian is essentially self adjoint on a domain a typical example is the pauli fierz model with spin 1 2 in nonrelativistic quantum electrodynamics for arbitrary coupling constants

remarks on the lower and upper solutions method for second and third order periodic boundary value problems we study the solvability and the approximation of the solutions by monotone iteration of the second and third order periodic boundary value problems bvps u cu t u u 0 u email protected u 0 u email protected and email protected au bu t u u 0 u email protected u 0 u email protected u 0 u email protected in the presence of lower and upper solutions which may not satisfy the usual ordering condition to this end some maximum and antimaximum principles are stated and proved for certain linear differential operators naturally associated with the previously stated problems

on the absence of eigenvectors of hamiltonians in a class of massless quantum field models without infrared cutoff a class of models of quantized massless bose fields called the generalized spin boson model a arai and m hirokawa j funct anal 151 1997 455 503 is considered theorems on the absence of ground states and the other eigenvectors of the model without infrared cutoff but with ultraviolet cutoff are established with conditions in terms of correlation functions for some operators

multiplicity of ground states in quantum field models applications of asymptotic fields abstract ground states of hamiltonian h of quantum field models are investigated the infimum of the spectrum of h is in the edge of its essential spectrum by means of the asymptotic field theory we give a necessary and sufficient condition for that the expectation value of the number operator of ground states is finite from which we give an upper bound of the multiplicity of ground states of h typical examples are massless gsb models and the pauli fierz model with spin 1 2

real polynomial diffeomorphisms with maximal entropy tangencies like the henon diffeomorphisms of r2 the quadratic maps of r have also played a central role in the field of dynamical systems these two families of dynamical systems fit together naturally which raises the question of the extent to which the dynamics is similar one difference is that our knowledge of these quadratic maps is much more thorough than our knowledge of these quadratic diffeomorphisms substantial progress in the theory of quadratic maps has led to a rather complete theoretical picture of their dynamics and an understanding of how the dynamics varies with the parameter despite significant recent progress in the theory of henon diffeomorphisms due to benedicks and carleson and many others there are still many phenomena that are not nearly so well understood in this two dimensional setting as they are for quadratic maps one phenomenon which illustrates the difference in the extent of our knowledge in dimensions one and two is the dependence of the complexity of the system on parameters in one dimension the nature of this dependence is understood and the answer is summarized by the principle of monotonicity loosely stated this is the assertion that the complexity of fa does not

enhanced binding of an n particle system interacting with a scalar bose field i an enhanced binding of an n particle system linearly coupled to a scalar bose field is investigated where n 2 it is not assumed that this system has a ground state for a zero coupling it is shown however that there exists a ground state for sufficiently large values of a coupling constant basic ingredients of the proof are a weak coupling limit and a modified hvz theorem

existence and construction of resonances for atoms coupled to the quantized radiation field abstract the processes of emission and absorption of photons by atoms can be rigorously understood in the low energy limit if we neglect the creation and annihilation of electrons they are related to eigenvalues of the atomic hamiltonian that are embedded in the continuous spectrum of the free hamiltonian of the electromagnetic field the mathematical analysis of the perturbation of these eigenvalues due to the electromagnetic interaction relies on a complex deformation technique relating the original hamiltonian to a non selfadjoint operator we develop a new technique to analyze the spectrum of operators used in non relativistic quantum electrodynamics our method can be applied to prove most of the results that previously required an involved renormalization group construction we use analytic perturbation theory of operators in hilbert spaces instead more precisely we extend the multi scale analysis introduced by one of us in 2003 15 which was used so far only for the study of selfadjoint operators to non selfadjoint operators compared to the selfadjoint case see for example 3 the analysis of these non selfadjoint operators is more difficult because we cannot make use of the functional calculus spectral theorem and the min max principle in some crucial estimates

scattering theory for a class of fermionic pauli fierz models abstract the scattering theory for a class of fermionic pauli fierz models is considered we give a proof of the asymptotic completeness of the dynamics in the case of massive fermions the result applied to the hamiltonian of a quantized spin 1 2 dirac particle interacting with an external field through a cutoff yukawa interaction and to the hamiltonian of a system of finitely many confined particles coupled to a fermionic field with a quadratic interaction