h asymptotic stability by fixed point in neutral nonlinear differential equations with delay abstract we use fixed point theory to obtain h asymptotic stability results about the zero solution of nonlinear differential equations with functional delay neither boundedness of the deviations nor a fixed sign on the coefficients functions are asked an explicit estimation of the solutions is obtained and a necessary and sufficient condition is determined classical results are improved and generalized

variationally stable differential systems let d e ir be a domain 0 e d and z a cc where a is a fixed real number let f c z x d if f t 0 0 consider the nonlinear system x f t x 1 and the perturbation g e c z x d rn such that for suitable functions ai 0 ii e c z i 1 p oi e c 0 co i 1 p nondecreasing and positive functions on 0 00 such that l w l o 11 and o oi are nondecreasing functions on 0 co we study the stability of the perturbed system y f t y g c yl knowing the stability of the variational systems 3 z fx t x t to xo z 4 where x t to x0 is the solution of 1 passing through the point to x0 e z x d thus we prove that if system 4 hence 1 is uniformly lipschitz stable see 3 then 3 is uniformly stable see theorem l and if q u uai clib l

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 ground state energy of the strong coupling polaron the polaron has been of interest in condensed matter theory and field theory for about half a century especially the limit of large coupling constant a it was not until 1983 however that a proof of the asymptotic formula for the ground state energy was finally given by using difficult arguments involving the large deviation theory of path integrals here we derive the same asymptotic result e 0 c 2 and with explicit error bounds by simple rigorous methods applied directly to the hamiltonian our method is easily generalizable to other settings e g the excitonic and magnetic polarons

asymptotic behavior of solutions of second order differential equations with integrable coefficients the differential equation x a t f x 0 t 0 is considered under the condition that lim x fl a s ds exists and is finite and necessary and or sufficient conditions are given for this equation to have solutions which behave asymptotically like nontrivial linear functions cl c2t

essential spectrum of a self adjoint operator on an abstract hilbert space of fock type and applications to quantum field hamiltonians abstract we establish general theorems on locating the essential spectrum of a self adjoint operator of the form a i i d s hi on the tensor product h f b k of a hilbert space h and the abstract boson fock space f b k over a hilbert space k where a is a self adjoint operator on h bounded from below d s is the second quantization of a nonnegative self adjoint operator s on k and hi is a symmetric operator on h f b k we then apply the theorems to the generalized spin boson model by a arai and m hirokawa 1997 j funct anal 151 455 503 and a general class of models of quantum particles coupled to a bose field including the pauli fierz model in nonrelativistic quantum electrodynamics

monotone positive solutions of second order nonlinear differential equations we obtain an existence theorem for monotone positive solutions of nonlinear second order ordinary differential equations by using the schauder tikhonov fixed point theorem the result can also be applied to prove the existence of positive solutions of certain semilinear elliptic equations in r n n greater than or equal to 3

an expression of the ground state energy of the spin boson model abstract an expression of the ground state energyesbof the spin boson hamiltonianhsbis considered the expression in the cases of both massive and massless bosons is given by a nonperturbative method by using the expression we show a necessary and sufficient condition with respect to a parameterg 1 0 such that a formula withgattains toesb

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

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

the asymptotic estimates of solutions of difference equations abstract the asymptotic estimate of the solution y t of linear difference equations with almost constant coefficients and the condition of asymptotic equivalence between y t and y 0 t are given where y 0 t is the corresponding solution of linear equations with constant coefficients a brief and elementary proof of the results is presented the method of proof carries over to differential equations in which some similar results are stated

limit point type solutions of nonlinear differential equations abstract we are concerned with the nonexistence of l 2 solutions of a nonlinear differential equation x a t x f t x by applying technique similar to that exploited by hallam siam j appl math 19 1970 430 439 for the study of asymptotic behavior of solutions of this equation we establish nonexistence of solutions from the class l 2 t 0 under milder conditions on the function a t which as the examples show can be even square integrable therefore the equation under consideration can be classified as of limit point type at infinity in the sense of the definition introduced by graef and spikes nonlinear anal 7 1983 851 871 we compare our results to those reported in the literature and show how they can be extended to third order nonlinear differential equations

convex polyominoes and heaps of segments builds a one to one correspondence between some polyominoes called parallelogram polyominoes and some heaps of segments then the author derives explicit expressions for the generating function of convex polyominoes according to their height width and area the author also enumerates two subsets of convex polyominoes namely the directed and convex polyominoes and the parallelogram polyominoes according to these three parameters generating functions of polyominoes having a fixed width are also studied

from object grammars to eco systems in this paper we make a comparison between two methods for the enumeration of combinatorial objects namely the eco method and object grammars both based on a recursive description for the examined class of objects in particular we study the problem of passing from an object grammar to an equivalent eco system first we solve this problem for any unidimensional unambiguous and complete object grammar with any linear parameter then we treat the more complex cases of q linear parameters and of multidimensional object grammars giving some explanatory examples in particular we determine a new eco system for the class of directed convex polyominoes

absence of ground state for the nelson model on static space times abstract we consider the nelson model on some static space times and investigate the problem of absence of a ground state nelson models with variable coefficients arise when one replaces in the usual nelson model the flat minkowski metric by a static metric allowing also the boson mass to depend on position we investigate the absence of a ground state of the hamiltonian in the presence of the infrared problem i e assuming that the boson mass m x tends to 0 at spatial infinity using path space techniques we show that if m x c x at infinity for some c 0 and 1 then the nelson hamiltonian has no ground state

positive solutions of nonlinear differential equations with prescribed decay of the first derivative an existence and uniqueness result for bounded positive solutions x t of the equation u f t u u 0 t greater than or equal to t 0 greater than or equal to 0 is established by means of the banach contraction principle for such a solution it is shown that alpha t less than or equal to x t less than or equal to beta t t greater than or equal to t 0 where alpha beta are given nonnegative continuous functions which are integrable over t 0 infinity the result complements others known in the literature

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

the zeros of basic bessel functions the functions jv ax x and associated orthogonal polynomials abstract the modified lommel polynomials are generalized and their orthogonality relation is obtained as a byproduct we prove that the non zero roots of jv ax x are real simple and interlace with those of jv ax 1 x the q lommel polynomials are shown to play in the theory of basic bessel function the role played by the lommel polynomials in the bessel function theory the q lommel polynomials are proved to be orthogonal with respect to a purely discrete measure with bounded support this is then used to prove that the prositive zeros of x j 2 v x q are real simple and interlace with the zeros of x 1j 2 v 1 x q when 1 we also establish the complete monotonicity of 1x 1 2 j v 1 2 i x q j v 2 i x q

spectral and scattering theory of spatially cut off p ϕ 2 hamiltonians we study spatially cut off p 2 hamiltonians we show the local finiteness of the pure point spectrum outside of thresholds the limiting absorption principle and asymptotic completeness of scattering for such hamiltonians our results imply the absence of singular continuous spectrum

on the asymptotic behavior of solutions of certain second order nonlinear differential equations on considere l equation differentielle py qy g t y y ou p p t est une fonction positive continument differentiable sur i a a telle que p a 1 q q t est une fonction continue sur i a et g g t u v est une fonction continue sur i a r 2 qui verifie certaines conditions

linear differential systems with conditionally integrable coefficients abstract for perturbations conditionally integrable levinson s theorems are obtained and applied to almost constant systems

renormalization group analysis of spectral problems in quantum field theory abstract in this paper we present a self contained and detailed exposition of the new renormalization group technique proposed in 1 2 its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator the partition function or correlation functions we apply this renormalization transformation to a hamiltonian describing the physics of an atom interacting with the quantized electromagnetic field and we prove that excited atomic states turn into resonances when the coupling between electrons and field is nonvanishing

spectral theory of pauli fierz operators abstract we study spectral properties of pauli fierz operators which are commonly used to describe the interaction of a small quantum system with a bosonic free field we give precise estimates of the location and multiplicity of the singular spectrum of such operators applications of these estimates which will be discussed elsewhere concern spectral and ergodic theory of non relativistic qed our proof has two ingredients the feshbach method which is developed in an abstract framework and mourre theory applied to the operator restricted to the sector orthogonal to the vacuum

an asymptotic analysis and its application to the nonrelativistic limit of the pauli fierz and a spin boson model an abstract asymptotic theory of a family of self adjoint operators h 0 acting in the tensor product of two hilbert spaces is presented and it is applied to the nonrelativistic limit of the pauli fierz model in quantum electrodynamics and of a spin boson model it is proven that the resolvent of h converges strongly as and the limit is a pseudoresolvent which defines an effective operator of h at as corollaries of this result some limit theorems for h are obtained including a theorem on spectral concentration an asymptotic estimate of the infimum of the spectrum the ground state energy of h is also given the application of the abstract theory to the above models yields some new rigorous results for them

generalization of a retarded gronwall like inequality and its applications abstract this paper generalizes a lipovan s result of gronwall like inequalities j math anal appl 252 2000 389 to a new type of retarded inequalities which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals from our result bihari s result and pinto s result can be deduced as some special cases our result can be applied to give the global existence and estimate solutions we also apply it to improve an estimation for almost periodicity of an invariant manifold

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

directed compact percolation near a wall iii exact results for the mean length and number of contacts existing exact results for the percolation probability and mean cluster size for compact percolation near a dry wall are extended to the mean cluster length and the mean number of contacts with the wall the results are derived from our previous work on vesicles near an attractive wall and involve elliptic integrals as opposed to the simple rational forms found for the percolation probability and cluster size below the results for the cluster length satisfy previously conjectured differential equations a closed expression is conjectured for the mean size above in terms of a hypergeometric function

asymptotic behavior of solutions of second order nonlinear difference equations 0 2 j 1 b n u n g n y y h 1 1 or equivalently e y a0 by g n y ey where e is the operator defined by ey n y n 1 2 let z z2 be a fundamental system of solutions of the linear difference equation lz e2z f aez bz 0 3 satisfying z o 1 z l 0 zz o 0 22 l 1 4 we will prove that there exists a ball b 0 p c r2 such that if y o y 1 e b 0 p then the solution y of equation 1 is defined on all n and for n co it satisfies

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

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

lipschitz stability of nonlinear systems of differential equations in s the authors introduced the notion of lipschitz stability in differential equations this notion lies somewhere between uniform stability on one side and the notions of asymptotic stability in variation 3 and uniform stability in variation 4 on the other side however lipschitz stability is new only as a nonliner phenomenon since it coincides with uniform stability in linear systems s an important feature of lipschitz stability is that unlike uniform stability the linearized system inherits the property of lipschitz stability from the original nonlinear system 5 in this paper we pursue the study of lipschitz stability that started in s using essentially the techniques of liapunov functions then we give sufficient conditions for the lipschitz stability of certain nonlinearly pertur bed nonlinear systems such systems include among other equations certain integrodifferential and functional differential equations then we give an example which can be investigated successfully using our results but cannot be handled by any previous techniques or results 9

nondegeneracy of the ground state for nonrelativistic lee model in the present work we first briefly sketch the construction of the nonrelativistic lee model on riemannian manifolds introduced in our previous works in this approach the renormalized resolvent of the system is expressed in terms of a well defined operator called the principal operator so as to obtain a finite formulation then we show that the ground state of the nonrelativistic lee model on compact riemannian manifolds is nondegenerate using the explicit expression of the principal operator that we obtained this is achieved by combining heat kernel methods with positivity improving semi group approach and then applying these tools directly to the principal operator rather than the hamiltonian without using cut offs

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

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

generating functions for classes of self avoiding polygons on the square lattice abstract a transfer matrix technique is presented which allows the explicit calculation of the generating functions for those self avoiding polygons on the square lattice which have a perimeter equal to or two larger than the perimeter of their bounding rectangles

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

on an involution on dyck paths les derniers resultats concernant l enumeration des polyamines parallelogrammes 4 5 8 9 11 14 selon l aire et le nombre de colonnes font apparaitre des fonctions generatrices s exprimant comme rapport de q analogues de fonctions de bessel dans 13 deux interpretations combinatoires de ces memes fonctions sont donnees d une part en termes de multiparcours d arbres et d autre part en termes de multichaines de chemins de dyck a l aide d une involution sur l ensemble d des chemins de dyck d une bijection entre chemins et multiparcours d arbres et d une bijection entre chemins et multichaines nous donnons dans ce papier une preuve bijective de l equivalence de ces deux interpretations

asymptotic representation of solutions of linear second order difference equations abstract using dichotomy conditions we obtain asymptotic formulae for the solutions of perturbed linear second order difference equations

l2 perturbation of a linear delay differential equation abstract in this paper we present an asymptotic formula for the solutions of the nonautonomous delay differential equation t a 0 a t x t b 0 b t x t assuming that it is an l 2 perturbation of the autonomous equation t a 0 x t b 0 x t our theorem is an essential generalization of the corresponding result in the classical book by r bellman and k l cooke the proof is based on a new result on the asymptotic constancy of the solutions of a linear delay equation

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

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

asymptotic behavior of the solutions of second order difference equations abstract the asymptotic behavior of the solutions of the second order difference equation 2 x n n x n is considered

ground states of the massless derezinski gerard model we consider the massless derezinski gerard model introduced by derezinski and gerard in 1999 we give a sufficient condition for the existence of a ground state of the massless derezinski gerard model without the assumption that the hamiltonian of particles has compact resolvent

existence of resonances for the spin boson model with critical coupling function abstract a two level atom coupled to the quantized radiation field is studied in the physical relevant situation the coupling function modeling the interaction between the two components behaves like k 1 2 as the photon momentum tends to zero this behavior is referred to as critical as it constitutes a borderline case previous results on non existence state that in the general case neither a ground state nor a resonance exists hasler and herbst 2011 have shown however that a ground state does exist if the absence of self interactions is assumed bach ballesteros konenberg and menrath 2017 have then explicitly constructed the ground state this specific case using the multiscale analysis known as pizzo s method pizzo 2003 in the present paper a resonance eigenvalue of the complex deformed hamiltonian is constructed by applying an inductive multiscale approach developed by bach ballesteros and pizzo 2013 2017 thus showing the existence of resonances in the critical case although the proof does not mainly rely on spectral renormalization some techniques enter in the form of a suitable feshbach schur map to control the exponential decay of the terms in the neumann series expansions overall the interaction terms obtained only involve second powers of the creation and annihilation operators and thus allow for a simpler proof

a dynamical systems result on asymptotic integration of linear differential systems abstract we are interested in the asymptotic integration of linear differential systems of the form x t r t x where is diagonal and r l p t 0 for p 1 2 our dichotomy condition is in terms of the spectrum of the omega limit set our results include examples that are not covered by the hartman wintner theorem

object grammars and bijections a new systematic approach for the specification of bijections between sets of combinatorial objects is presented it is based on the notion of object grammars object grammars give recursive descriptions of objects and generalize context free grammars the study of a particular substitution in these object grammars confirms once more the key role of dyck words in the domain of enumerative and bijective combinatorics

asymptotic behavior of nonlinear difference systems in this paper we study a general variational stability introduced mainly for weakly stable difference systems moreover we obtain asymptotic formulae for these systems which state new results about asymptotic behavior for perturbed systems under general hypotheses

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

integral inequalities of gronwall bellman bihari type and asymptotic behavior of certain second order nonlinear differential equations abstract several integral inequalities similar to gronwall bellmann bihari inequalities are obtained these inequalities are used to discuss the asymptotic behavior of certain second order nonlinear differential equations

existence of ground state eigenvalues for the spin boson model with critical infrared divergence and multiscale analysis abstract a two level atom coupled to the radiation field is studied first principles in physics suggest that the coupling function representing the interaction between the atom and the radiation field behaves like k 1 2 as the photon momentum k tends to zero previous results on non existence of ground state eigenvalues suggest that in the most general case binding does not occur in the spin boson model i e the minimal energy of the atom photon system is not an eigenvalue of the energy operator hasler and herbst have shown 12 however that under the additional hypothesis that the coupling function be off diagonal which is customary to assume binding does indeed occur in this paper an alternative proof of binding in case of off diagonal coupling is given i e it is proven that if the coupling function is off diagonal the ground state energy of the spin boson model is an eigenvalue of the hamiltonian we develop a multiscale method that can be applied in the situation we study with the help of a key symmetry operator which we use to demonstrate that the most singular terms appearing in the multiscale analysis vanish

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

algebraic languages and polyominoes enumeration abstract in this paper the use of algebraic languages theory in solving an open problem in combinatorics is shown by constructing a bijection between convex polyominoes and words of an algebraic language and by solving the corresponding algebraic system we prove that the number of convex polyominoes with perimeter 2n 8 is 2n 11 4 n 4 2n 1 2n n

a generalized retarded gronwall like inequality in two variables and applications to bvp abstract in this paper we establish a generalized retarded integral inequality of gronwall like type in two variables which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals without assumption of monotonicity using our result we can solve both the integral inequality in w s cheung some new nonlinear inequalities and applications to boundary value problems nonlinear anal 64 2006 2112 2128 and the one in r p agarwal s deng w zhang generalization of a retarded gronwall like inequality and its applications appl math comput 165 2005 599 612 we apply our result to a boundary value problem of a partial differential equation for boundedness uniqueness and continuous dependence

on the asymptotic behavior of solutions of higher order nonlinear differential equations on considere l equation differentielle py n 1 i 1 n 1 b i t y i 1 g t y y y n 1 ou p p t est une fonction positive et continument derivable sur i a a telle que p a 1 h i h i t sont continues sur i a et g est continu sur i a r n verifiant quelques autres conditions en considerant un systeme fondamental de solutions on caracterise certaines solutions definies sur l intervalle i a entier

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 summation for second order finite difference systems abstract conditional summable hypotheses are given to obtain asymptotic summation of some 2 2 difference systems we obtain asymptotic formulae of the solutions of a perturbed system knowing the recessive and dominant solutions of the unperturbed system the casoratian is generally nonconstant and nonconvergent and it appears in the asymptotic formulas

spectral theory of massless pauli fierz models we study the spectral theory of massless pauli fierz models using an extension of the mourre method we prove the local finiteness of point spectrum and a limiting absorption principle away from the eigenvalues for an arbitrary coupling constant in addition we show that the expectation value of the number operator is finite on all eigenvectors

attribute grammars are useful for combinatorics abstract the purpose of this paper is to show the use of attribute grammars in solving some combinatorics problems for example we give the generating function for the shape of skew ferrers diagrams according to the number of cells and the number of columns this result is new and proves that skew ferrer diagrams are related to new basic bessel functions

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

statistical mechanics of directed models of polymers in the square lattice directed square lattice models of polymers and vesicles have received considerable attention in the recent mathematical and physical sciences literature these are idealized geometric directed lattice models introduced to study phase behaviour in polymers and include dyck paths partially directed paths directed trees and directed vesicles models directed models are closely related to models studied in the combinatorics literature and are often exactly solvable they are also simplified versions of a number of statistical mechanics models including the self avoiding walk lattice animals and lattice vesicles the exchange of approaches and ideas between statistical mechanics and combinatorics have considerably advanced the description and understanding of directed lattice models and this will be explored in this review the combinatorial nature of directed lattice path models makes a study using generating function approaches most natural in contrast the statistical mechanics approach would introduce partition functions and free energies and then investigate these using the general framework of critical phenomena generating function and statistical mechanics approaches are closely related for example questions regarding the limiting free energy may be approached by considering the radius of convergence of a generating function and the scaling properties of thermodynamic quantities are related to the asymptotic properties of the generating function in this review the methods for obtaining generating functions and determining free energies in directed lattice path models of linear polymers is presented these methods include decomposition methods leading to functional recursions as well as the temperley method that is implemented by creating a combinatorial object one slice at a time a constant term formulation of the generating function will also be reviewed the thermodynamic features and critical behaviour in models of directed paths may be informative about the underlying properties that determine phase diagrams for wider classes of models including physical models of polymers of particular interest are adsorption and collapse transitions in models of polymers and copolymers the properties of thermodynamic quantities in those models are described by tricritical scaling this is reviewed for directed path models and the generating function approaches can be used to apply tricritical scaling to models of adsorbing inflating and collapsing directed lattice paths critical exponents for a variety of models can be obtained in this manner and with it a better understanding and a classification of the models

strong uniqueness for both dirichlet operators and stochastic dynamics to gibbs measures on a path space with exponential interactions abstract we prove l p uniqueness of dirichlet operators for gibbs measures on the path space c r r d associated with exponential type interactions in infinite volume by extending an spde approach presented in previous work by the last two named authors we also give an spde characterization of the corresponding dynamics in particular for the first time we prove existence and uniqueness of a strong solution for the spde though the self interaction potential is not assumed to be differentiable hence the drift is possibly discontinuous as examples to which our results apply we mention the stochastic quantization of p 1 exp 1 and trigonometric quantum fields in infinite volume in particular our results imply essential self adjointness of the generator of the stochastic dynamics for these models finally as an application of the strong uniqueness result for the spde we prove some functional inequalities for diffusion semigroups generated by the above dirichlet operators

on the boundedness and stability of the solutions of semilinear differential equations abstract we study the global existence boundedness and stability of solutions of a class of semilinear differential equations the results obtained give very good estimates and an explicit radius of attraction

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

positive commutators fermi golden rule and the spectrum of zero temperature pauli fierz hamiltonians abstract we perform the spectral analysis of a zero temperature pauli fierz system for small coupling constants under the hypothesis of fermi golden rule we show that the embedded eigenvalues of the uncoupled system disappear and establish a limiting absorption principle above this level of energy we rely on a positive commutator approach introduced by skibsted and pursued by georgescu gerard moller we complete some results obtained so far by derezinski jaksic on one side and by bach frohlich sigal soffer on the other side

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

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

global existence of solutions with prescribed asymptotic behavior for second order nonlinear differential equations where the nonlinearity f satis9es certain monotonicity conditions speci9ed in the sequel we always assume without mentioning that the function f t u v is continuous in the domain d t u v t t0 u v r where t0 1 it is supposed also that all equations and inequalities involving t hold for t t0 unless otherwise speci9ed the symbols o and o have their standard meanings as t and we use the notation r def 0 eq 1 is often used for mathematical modelling of various physical chemical and biological systems and attracts constant interest of researchers a great deal of papers published during the last three decades are concerned with local and global existence of solutions of eq 1 and its numerous particular cases uniqueness of solutions

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

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

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

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