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

a time dependent theory of quantum resonances in this paper we further develop a general theory of metastable states resulting from perturbation of unstable eigenvalues we apply this theory to many body schrouml dinger operators and to the problem of quasiclassical tunneling

existence and roughness of exponential dichotomies of linear dynamic equations on time scales in this paper we define the exponential dichotomy of linear dynamic equations on time scales then we present perturbation theorems on the roughness of exponential dichotomy and develop several explicit sufficient criteria for linear dynamic equations to have an exponential dichotomy as applications of the criteria of exponential dichotomy we derive some new sufficient conditions for the existence of periodic solutions of semi linear dynamic equations and nonlinear dynamic equations on time scales

nonautonomous impulsive systems with unbounded nonlinear terms in fenner and pinto 1999 23 proved that if the linear impulsive system satisfies the is condition see definition 2 1 with bounded nonlinear term f t x then the perturbed nonlinear impulsive system has a unique bounded solution see theorem a the method used to prove theorem a cannot be applied to the unbounded case in this paper we prove that if f t x e t m then the perturbed nonlinear impulsive system has a unique solution

dichotomies and asymptotic solutions of nonlinear differential systems the problem of asymptotic equivalence between two systems of ordinary differential equations has been studied by many authors l 5 it was first used by levinson in his theorem on asymptotic integration 6 71 in l 51 this problem is studied when system 1 2 is stable a t is constant and g t y b t is integrable in 3 theorems 5 111 it is assumed that the linear system 1 2 has an ordinary dichotomy and the nonlinear term g t y is lipschitz continuous in 2 the previous results are extended to nonlinear perturbations g t y which are not lipschitz continuous such that g t y l i r t y where r t y is monotone in y we emphasize that in all these works only the asymptotic equivalence between bounded solutions is established nothing is said about a possible correspondence between unbounded solutions of 1 1 and 1 2 in this paper we suppose that the nonlinear term g t y satisfies some integrability conditions that we summarize in hypothesis h2 these conditions permit one to consider systems with large nonlinearities for example equations where g t y is an oscillatory function of the variable y no lipschitz conditions or monotonicity properties are required of g t y however we will use dichotomic properties of system 1 2

gibbs measures on brownian currents motivated by applications to quantum field theory we consider gibbs measures for which the reference measure is wiener measure and the interaction is given by a double stochastic integral and a pinning external potential in order properly to characterize these measures through dlr equations we are led to lift wiener measure and other objects to a space of configurations where the basic observables are not only the position of the particle at all times but also the work done by test vector fields we prove existence and basic properties of such gibbs measures in the small coupling regime by means of cluster expansion

polynomial growth rates abstract we consider linear equations v a t v with a polynomial asymptotic behavior that can be stable unstable and central we show that this behavior is exhibited by a large class of differential equations by giving necessary and sufficient conditions in terms of generalized polynomial lyapunov exponents for the existence of polynomial behavior in particular any linear equation in block form in a finite dimensional space with three blocks having polynomial lyapunov exponents respectively negative positive and zero has a nonuniform version of polynomial trichotomy which corresponds to the usual notion of trichotomy but now with polynomial growth rates we also obtain sharp bounds for the constants in the notion of polynomial trichotomy in addition we establish the persistence under sufficiently small nonlinear perturbations of the stability of a nonuniform polynomial contraction

structural stability of linear difference equations in hilbert space abstract in this note we prove that linear difference equations in hilbert space are structurally stable if and only if they have an exponential dichotomy

chain rule and invariance principle on measure chains in this note we prove a chain rule for mappings on abstract measure chains and apply our result to deduce an invariance principle for non autonomous dynamic equations

weighted convergent and bounded solutions of difference systems abstract using a summable dichotomy and fixed point theorems we prove the existence of convergent and bounded solutions of nonlinear difference systems

extremal values of the sojourn time consider a self adjoint operator h on a separable hilbert space with non trivial absolutely continuous component we study the general properties of the real valued functional which in quantum mechanics represents the sojourn time or life time of an initial state we characterize the critical points of the sojourn time x of the operator multiplication by x in and prove that it attains a global maximum in the unit sphere of the sobolev space

existence and roughness of the exponential dichotomy for skew product semiflow in banach spaces abstract in this paper we introduce a concept of exponential dichotomy for skew product semiflow in infinite dimensional banach spaces which is an extension of the classic concept for evolution operators this concept is used to study the roughness property of the skew product semiflow also we introduce the concept of discrete skew product and give a necessary and sufficient condition for this discrete skew product to have a discrete dichotomy after that we give necessary and sufficient conditions for the existence of exponential dichotomy for skew product semiflow moreover we prove that the exponential dichotomy for skew product semiflow is not destroyed by small perturbation finally we apply these results to parabolic partial differential equations and functional differential equations

exponential dichotomy roughness and structural stability for evolution families without bounded growth and decay abstract we show that if u t s is an exponentially dichotomic evolution operator then the unique solution of the volterra equation v t s u t s s t u t b v s d is also an exponentially dichotomic evolution operator for small b t as a consequence we prove that any exponentially dichotomic evolution family is structurally stable we improve the previous results showing that the condition of bounded growth and decay present everywhere in the existing literature can be completely removed

lp boundedness properties for volterra difference equations this work deals with l p boundedness of solutions of a class of volterra functional difference equation the effect of different kinds of the nonhomogeneous term lipschitz local lipschitz completely continuous are taken into consideration

a grobman hartman theorem for nonuniformly hyperbolic dynamics abstract we establish a version of the grobman hartman theorem in banach spaces for nonuniformly hyperbolic dynamics we also consider the case of sequences of maps which corresponds to a nonautonomous dynamics with discrete time more precisely we consider sequences of lipschitz maps a m f m such that the linear parts a m admit a nonuniform exponential dichotomy and we establish the existence of a unique sequence of topological conjugacies between the maps a m f m and a m furthermore we show that the conjugacies are holder continuous with holder exponent determined by the ratios of lyapunov exponents with the same sign to the best of our knowledge this statement appeared nowhere before in the published literature even in the particular case of uniform exponential dichotomies although some experts claim that it is well known in this case we are also interested in the dependence of the conjugacies on the perturbations f m we show that it is holder continuous with the same holder exponent as the one for the conjugacies we emphasize that the additional work required to consider the case of nonuniform exponential dichotomies is substantial in particular we need to consider several additional lyapunov norms

asymptotic integration of ordinary different systems an extension and a new proof of hartman and wintner s theorem on asymptotic integration are given

a topological classification of linear differential equations on banach spaces abstract in the present work we give necessary and sufficient conditions for linear differential non autonomous equations on a banach space that satisfy exponential dichotomy to be topologically equivalent we shall also prove that such equations are structurally stable

shape resonances a comparison between parabolic and flat bottom potentials abstract sharp exponential bounds are obtained for resonance states and complex eigenvalues z for the schrodinger equation h d 2 dx 2 v z on 0 in a compact region v x is either a square well or a parabolic potential as x v x v v m x for some 0 the three cases v v 0 and 0 v v m are considered results are used to study the stratified wave propagator k c 2 y x y depending on the shape of the velocity function c s resonances of k are shown to exist

structured stability radii and exponential stability tests for volterra difference systems uniform exponential ue stability of linear difference equations with infinite delay is studied using the notions of a stability radius and a phase space the state space x is an abstract banach space we work with non fading phase spaces c 0 z x and z x and with exponentially fading phase spaces of the p and c 0 types for equations of the convolution type several criteria of ue stability are obtained in terms of the z transform email protected z of the convolution kernel k in terms of the input state operator and of the resolvent fundamental matrix these criteria do not impose additional positivity or compactness assumptions on coefficients k j time varying non convolution difference equations are studied via structured ue stability radii r t of convolution equations these radii correspond to a feedback scheme with delayed output and time varying disturbances we also consider stability radii r c associated with a time invariant disturbance operator unstructured stability radii and stability radii corresponding to delayed feedback for all these types of stability radii two sided estimates are obtained the estimates from above are given in terms of the z transform email protected z the estimate from below via the norm of the input output operator these estimates turn into explicit formulas if the state space x is hilbert or if disturbances are time invariant the results on stability radii are applied to obtain various exponential stability tests for non convolution equations several examples are provided

a unified floquet theory for discrete continuous and hybrid periodic linear systems abstract in this paper we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts e g hybrid dynamical systems we develop a comprehensive floquet theory including lyapunov transformations and their various stability preserving properties a unified floquet theorem which establishes a canonical floquet decomposition on time scales in terms of the generalized exponential function and use these results to study homogeneous as well as nonhomogeneous periodic problems furthermore we explore the connection between floquet multipliers and floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales finally we show this unified floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated but time varying system s poles in the complex plane we include several examples to show the utility of this theory

topological equivalence and structural stability for linear difference equations abstract in this paper first we prove that if two linear difference equations with invertible coefficient matrices are topologically equivalent and one of them has bounded coefficient matrix together with its inverse then the coefficient matrix of the other equation is also bounded together with its inverse we also prove that if a linear difference equation with bounded and invertible coefficient matrix is structurally stable then the inverse of the coefficient matrix is also bounded

exponential dichotomies and homoclinic orbits in functional differential equations abstract suppose an autonomous functional differential equation has an orbit which is homoclinic to a hyperbolic equilibrium point the purpose of this paper is to give a procedure for determining the behavior of the solutions near of a functional differential equation which is a nonautonomous periodic perturbation of the original one the procedure uses exponential dichotomies and the fredholm alternative it is also shown that any smooth function p t defined on the reals which approaches zero monotonically as t is the solution of a scalar functional differential equation and generates an orbit homoclinic to zero examples illustrating the results are also given

nonuniform h k μ ν dichotomy with applications to nonautonomous dynamical systems the paper develops and studies a very general notion of dichotomy referred to as nonuniform h k dichotomy the new notion contains as special cases most versions of dichotomy existing in the literature the paper then provides corresponding new versions of robustness hartman grobman theorem and stable manifold theorem for nonautonomous dynamical systems in banach spaces in term of the nonuniform h k dichotomy

sojourn time for rank one perturbations we consider a self adjoint purely absolutely continuous operator m let p be a rank one operator pu u such that for 0 h 0 m 0p has a simple eigenvalue e0 embedded in its absolutely continuous spectrum with corresponding eigenvector let h be a rank one perturbation of the operator h 0 namely h m 0 p under suitable conditions the operator h has no point spectrum in a neighborhood of e0 for small here we study the evolution of the state under the hamiltonian h in particular we obtain explicit estimates for its sojourn time e ih t 2dt by perturbation theory we prove that is finite for 0 and that for small it is of order 2 finally by using an analytic deformation technique we estimate the sojourn time for the friedrichs model in rn

convergent solutions of linear functional difference equations in phase space by using weighted summable dichotomies and schauder s fixed point theorem we prove the existence of convergent solutions of linear functional difference equations we apply our result to volterra difference equations with infinite delay

the gibbs ensemble of a vortex filament we introduce a statistical ensemble for a single vortex filament of a three dimensional incompressible fluid the core of the vortex is modeled by a quite generic stochastic process we prove the existence of the partition function for both positive and a limited range of negative temperatures

cauchy matrix for linear almost periodic systems and some consequences abstract a novel method to construct the fundamental matrix for a linear almost periodic system is proposed provided that the diagonal terms satisfy an average separation condition and the off diagonal coefficients are l small the idea is to transform the system in a set of riccati type equations and use exponential dichotomy and its consequences it is shown that the method yields easy computation procedures with simple and direct conditions depending on the coefficients finally our result enables us to obtain i explicit almost periodic matrices q t q 1 t and q t which diagonalize the original system and ii sufficient conditions for the stability two illustrative examples are shown

exponential dichotomy and trichotomy for difference equations abstract in this paper a roughness theorem of exponential dichotomy and trichotomy of linear difference equations is proved it is also shown that if an almost periodic difference equation has an exponential dichotomy on a sufficiently long finite interval then it has one on

asymptotic integration of delay differential systems on considere des equations non autonomes qui sont asymptotiquement autonomes et on cherche des formules pour les solutions aux grandes valeurs de la variable independante

h k trichotomies and asymptotics of nonautonomous difference systems abstract discrete nonautonomous nonlinear systems possessing h k trichotomies are considered we specialize to the case of solutions moving inside and in a neighborhood of an invariant manifold of a special kind which we call h k hyperbolic the aulbach coppel knobloch transformation is then developed and used to state conditions for granting asymptotic equivalence of solutions our results generalize some of aulbach s results in which the manifold considered was made of stationary points and the system taken was autonomous

robustness of nonuniform μ ν dichotomies in banach spaces abstract this paper focuses on the linear perturbation of dynamical systems in banach spaces which corresponds to a nonautonomous dynamical system admitting a nonuniform dichotomy we establish the robustness or roughness of nonuniform dichotomies in the sense that the linear dynamical system persists under a sufficiently small linear perturbation

on the semigroup generated by the renormalized nelson hamiltonian abstract we consider the renormalized nelson model at a fixed total momentum p h ren p the hamiltonian h ren p is defined through an infinite energy renormalization we prove that e h ren p is positivity improving for all p r 3 and 0 in the fock representation

on admissible perturbations for exponential dichotomy in this work we obtain an optimal upper bound for exponential dichotomy roughness in infinite dimensional banach spaces unlike some previous works we do not assume bounded growth we consider linear non autonomous ordinary differential equations with bounded and unbounded coefficients

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

existence and robustness of nonuniform h k μ ν dichotomies for nonautonomous impulsive differential equations abstract for the linear nonautonomous impulsive differential equation we propose a new notion called the nonuniform h k dichotomy which extends the notion of dichotomy of uniform and nonuniform types and is related to the theory of nonuniform hyperbolicity we establish a sufficient criterion for the existence of the nonuniform h k dichotomy in terms of appropriate lyapunov exponents for the linear perturbation we develop the robustness or roughness of nonuniform h k dichotomies in the sense that a nonuniform h k dichotomy persists under a sufficiently small linear perturbation

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

reducibility of linear dynamic equations on measure chains the concepts of reducibility and kinematic similarity are of major significance in the theory of stability of linear differential and difference equations in this paper we generalize some fundamental results on reducibility from the finite dimensional differential equations context to dynamic equations on measure chains in arbitrary hilbert spaces in fact we derive sufficient conditions for dynamic equations to be kinematically similar to an equation with zero right hand side or to an equation in hermitian or block diagonal form

exponential dichotomies for dynamic equations on measure chains in this paper we introduce the notion of an exponential dichotomy for not necessarily invertible linear dynamic equations in banach spaces within the framework of the calculus on measure chains particularly this unies the corresponding theories for dierence and dierential equations we apply our approach to obtain results on perturbed systems

exponential dichotomy roughness on banach spaces in the present paper we extend existing results on exponential dichotomy roughness for linear ode systems to infinite dimensional banach space we give new conditions for the existence of exponential dichotomy roughness in infinite dimensional space and in the finite interval case we also improve previous results by indicating the exact values of the dichotomic constants of the perturbed equation

robustness of general dichotomies abstract for a linear equation v a t v we consider general dichotomies that may exhibit stable and unstable behaviors with respect to arbitrary asymptotic rates e c t for some function t this includes as a special case the usual exponential behavior when t t we also consider the general case of nonuniform exponential dichotomies we establish the robustness of the exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation persists under sufficiently small linear perturbations we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy

invariant manifolds with asymptotic phase this paper is concerned with invariant manifolds allowing a generic approach to solutions of the underlying autonomous differential system only this means that any solution approaching such an invariant manifold m ultimately behaves like a particular solution on j4 in the sense that the difference n t i l of the two solutions tends to 0 as t cq in other words only solutions on the stable manifold of m can approach m as f cc invariant manifolds with that property are said to have an asymptotic phase the simplest nontrivial example of an invariant manifold with asymptotic phase is the orbit of an isolated periodic solution whose floquet multipliers lie all but one in the open unit disc of the complex plane malkin 12 proved a manifold of stationary solutions to have an asymptotic phase if each of these solutions has all eigenvalues with nonpositive real parts and if in addition the number of eigenvalues on the imaginary axis equals the dimension of the manifold more generally a k dimensional manifold generated by a family of periodic solutions has an asymptotic phase if all but k floquet multipliers have modulus less than 1 see hale stokes 9 conditions for a torus to be asymptotically stable with asymptotic phase are given by samoilenko 13 and coppel 5 invariant manifolds without any specified structure are considered by fenichel 6 7 and in kirchgraber s note 111 moreover each center manifold has an asymptotic phase if the corresponding equilibrium is stable see carr al amood 4 and henry lo in any case the condition guaranteeing the existence of an asymptotic phase amounts to the fact that the decay rate of solutions toward the manifold is greater than the decay rate of the solutions within the manifold in each of the cases quoted so far the invariant manifold is asymptotically stable and thus the question arises whether an asymptotic phase can exist only in connection with asymptotic stability a negative answer to this question is given by the author in l 21 where hyperbolic manifolds of periodic solutions are proved to have an asymptotic phase in l the manifold is supposed to be compact while the noncompact case is treated in 2 under the restriction that the generating family of periodic solutions has an amplitude independent period recently hale massatt 8 extended the corresponding result for manifolds of stationary solutions to a certain class of partial differential equations in this paper we drop the periodicity assumption of the flow on the given manifold and prove the following result a compact invariant manifold has an asymptotic phase if the manifold is hyperbolic assumption hl and carries a flow parallel in the sense of our assumption h2 this includes most of the aforementioned results

the hartman wintner theorem for functional differential equations abstract in this paper we present a generalization of the hartman wintner theorem about the asymptotic behavior of the solutions of linear ordinary differential equations to functional differential equations

asymptotic integration of linear differential equations abstract we establish a new result on the asymptotic integration of linear differential systems of the form y r y where is diagonal and r is small more specifically we allow for different scales of asymptotic behaviour of the individual matrix elements of and r we also discuss an application to a self adjoint fourth order equation

roughness of h k dichotomies abstract in this paper we prove a roughness theorem for a class of dichotomies of linear systems of differential equations the so called h k dichotomies these include a great variety of dichotomies among them the ordinary and exponential dichotomies exponential ordinary dichotomies levinson dichotomies for diagonal systems etc we prove that under general uniform conditions upon the functions h and k these dichotomies are not destroyed by integrable perturbations this result generalizes the property of roughness of ordinary dichotomies under integrable perturbations

admissibility criteria for nonuniform dichotomic behavior of nonautonomous systems on the whole line abstract we give new criteria for nonuniform dichotomy of nonautonomous systems on the whole line in terms of admissibility relative to an integral equation in our approach the input space i r x is an intersection of spaces that can be successively minimized and the output space c r x can be one of some well known spaces of continuous functions using computational arguments we show that the admissibility of c r x i r x leads to a nonuniform exponential dichotomy we expose a complete analysis of the connections between admissibility and nonuniform dichotomy on the whole line and we also discuss several interesting consequences moreover we obtain the explicit expression of the growth rates for dichotomy in terms of the initial exponential growth and the norm of the input output operator finally we present a direct application of the main result in the case of evolution families which admit uniform exponential growth

asymptotic properties of solutions to nonautonomous volterra difference systems with infinite delay abstract we show that under certain conditions there is a one to one correspondence between weighted bounded solutions of a linear nonautonomous volterra difference system with infinite delay and its perturbed system under additional assumptions this correspondence is elevated to asymptotic equivalence on the other side we prove the existence of weighted convergent solutions

asymptotic behavior in volterra difference systems with unbounded delay we show that under certain conditions there is a one to one correspondence between weighted bounded solutions of a linear volterra difference equation with unbounded delay and its perturbation this correspondence is elevated to asymptotic equivalence under suitable conditions

asymptotic integration of a system resulting from the perturbation of an h system z systems were introduced in lo with the intention of obtaining results about stability for a weakly stable system at least weaker than those given by exponential stability under some perturbations in this paper we get asymptotic formulae for these systems which state new results about asymptotic integration for nonlinear systems under general hypotheses moreover the corresponding results for linear systems give new insight about levinson s theorem on asymptotic integration in this way using the h systems we get a uniform treatment for the concept of stability and we verify that several classical theorems of stability take a precise form by the asymptotic formulae in section 2 we recall the definition and some properties of an h system and we state two new theorems about asymptotic behavior of solutions of perturbed h systems in section 3 some general results about asymptotic integration are given these results extend those given in lo in section 4 we apply the technique to the linear case finally in section 5 we show examples and applications and we study the emden fowler equation which originally does not satisfy the general hypothesis but can be transformed into one where they are satisfied 2 h systems let us consider the differential system

on weak exponential stability of evolution operators in banach spaces abstract the aim of this paper is to give some characterizations for weak exponential stability properties of evolution operators in banach spaces variants for weak exponential stability of some well known results in uniform stability theory buse and niculescu 2009 1 daleckij and krein 1974 2 datko 1973 3 rolewicz 1986 7 stoica and megan 2009 8 are obtained

robustness of nonuniform exponential dichotomies in banach spaces we give conditions for the robustness of nonuniform exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation x a t x persists under a sufficiently small linear perturbation we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the angles between the stable and unstable subspaces our proofs exhibit implicitly the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions we emphasize that we do not need the notion of admissibility of bounded nonlinear perturbations we also obtain related robustness results in the case of nonuniform exponential contractions in addition we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time

on the linearization theorem of fenner and pinto this paper reports an improvement of the linearization theorem of fenner and pinto 1999 22 fenner and pinto presented a version of hartman s result they showed that there exists a one to one correspondence between solutions of the linear system and the nonlinear system moreover if h t x denotes the transformation then h t x x is uniformly bounded however no proof of the holder regularity of the transformation h t x appears in fenner and pinto 1999 22 the main objective in this paper is precisely to give a proof of the holder regularity of the transformation h t x namely we show that the conjugating function h t x in the hartman grobman theorem is always holder continuous and has holder continuous inverse moreover we weakened an important assumption in fenner and pinto 1999 22 fenner and pinto obtained the linearization theorem by setting that the whole linear system should satisfy is condition in this paper this assumption is reduced in fact it is enough to assume that the linear system partially satisfies is condition therefore we improve the linearization theorem of fenner and pinto

dichotomies and asymptotic formulas for the solutions of differential equations abstract the new concept of h k dichotomy is studied extensively and is used in asymptotic equivalence problems

ultraviolet renormalization of the nelson hamiltonian through functional integration starting from the n particle nelson hamiltonian defined by imposing an ultraviolet cutoff we perform ultraviolet renormalization by showing that in the ultraviolet cutoff limit a self adjoint operator exists after a logarithmically divergent term is subtracted from the original hamiltonian we obtain this term as the diagonal part of a pair interaction appearing in the density of a gibbs measure derived from the feynman kac representation of the hamiltonian also we show existence of a weak coupling limit of the renormalized hamiltonian and derive an effective yukawa interaction potential between the particles

asymptotic integration of nonautonomous delay differential systems we prove existence theorems and asymptotic formulas for the solutions of a class of delay differential equations with time state dependent lag

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

scaling limits with a removal of ultraviolet cutoffs for semi relativistic particles system coupled to a scalar bose field abstract the system of semi relativistic particles coupled to a scalar bose field is considered a scaled total hamiltonian for the system is a self adjoint operator on a tensor product of a square integrable function space and a boson fock space we consider the strong resolvent limit of a renormalized hamiltonian which is defined by subtracting a divergent term from the scaled total hamiltonian by an abstract scaling limit theory and a unitary transformation we derive the yukawa potential and coulomb potential as effective potentials

on roughness of exponential dichotomy we present new perturbation theorems on the roughness of exponential dichotomy which improve previous results the proofs given here are also much simpler compared with previous ones the new results provide significant improvements of existing results in the case where the operator a t is unbounded for t j this is precisely the situation that is of interest from the point of view of the applications in lobe dynamics the results are also valid in banach spaces and useful for general purposes

asymptotic behavior of solutions of poincare recurrence systems abstract sufficient conditions are given for the poincare recurrence system y m 1 a p m y m to have a solution such that y m m 1 o 1 v as m where is an eigenvalue of the constant matrix a and v is an associated eigenvector the summability conditions on p permit conditional convergence and the o 1 terms are specified precisely

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

about the behavior of solutions for volterra difference equations with infinite delay we study the existence of discrete almost automorphic solutions and asymptotic behavior for non linear volterra difference equations of convolution type with infinite delay where the nonlinear perturbation is considered not necessarily globally lipschitz the results are a consequence of application of different fixed point theorems namely the contraction principle the leray schauder the krasnoselkii and matkowski s fixed point technique

an asymptotic theory for nonlinear functional differential equations abstract the hartman wintner theorem on asymptotic integration is established for a certain class of functional differential equations with nonlinear perturbations by looking them as abstract ordinary differential equations results for delay differential equations are included in this study consequences and applications are also shown

an asymptotic theory for retarded functional difference equations using discrete dichotomies and krasnoselsky s theorem we obtain existence and asymptotic behavior of convergent solutions for retarded functional difference equations we also will get some global properties for the set of convergent solutions applications on volterra difference equations with infinite delay are shown

stable manifolds for nonuniform polynomial dichotomies we establish the existence of smooth stable manifolds in banach spaces for sufficiently small perturbations of a new type of dichotomy that we call nonuniform polynomial dichotomy this new dichotomy is more restrictive in the nonuniform part but allow the uniform part to obey a polynomial law instead of an exponential more restrictive law we consider two families of perturbations for one of the families we obtain local lipschitz stable manifolds and for the other family assuming more restrictive conditions on the perturbations and its derivatives we obtain c1 global stable manifolds finally we present an example of a family of nonuniform polynomial dichotomies and apply our results to obtain stable manifolds for some perturbations of this family

on uniform exponential stability for skew evolution semiflows on banach spaces the aim of this paper is to give several characterizations for the property of stability for skew evolution semiflows on banach spaces thus we obtain generalizations of some well known results due to barbashin datko and rolewicz in the case of evolution equations in banach spaces a unified treatment in the uniform setting is provided

a reduction theorem for systems of differential equations with impulse effect in a banach space a reduction theorem for systems of differential equations with impulse effect at fixed moments in a banach space is proven this result allows one to substantially reduce the given system to a much simpler one

asymptotic property of linear volterra difference systems by using the resolvent matrix and the comparison principle we investigate the asymptotic behavior of linear volterra difference systems

dichotomy and existence of periodic solutions of quasilinear functional differential equations abstract using krasnoselskii s fixed point theorem functional analysis methods and dichotomy theory we study the existence and uniqueness of the periodic solutions of integro differential equations with bounded and unbounded delays

on the evolution of resonance states we consider two body schrodinger operators with multiplicative exponentially decreasing and form compact potentials it is proved that the evolution of sharp cutoff approximations of a resonance function is outgoing and exponentially damped except for the choices of cutoff radii shown to be determined by outgoing asymptotics an explicit error estimate is given in terms of time variable resonance energy and width

on h k manifolds with asymptotic phase abstract we introduce the notion of h k manifolds and give conditions under which the property of being a manifold with asymptotic phase holds these conditions allow the construction of a transformation of the variational equation of a nonlinear nonautonomous systems with h k dichotomies into an almost decoupled quasilinear system the class of admissiblehandkfunctions is also briefly analyzed