approximate solution of general high order linear nonhomogeneous difference equations by means of taylor collocation method in this paper a taylor collocation method is developed to find an approximate solution of general high order linear nonhomogenous difference equations with variable coefficients under the mixed conditions the solution is obtained in terms of taylor polynomials about any point also examples are presented which illustrate the pertinent features of the method

bounded perturbations of the generalized random fields functional dlr equations corresponding to the bounded perturbations of sufficiently regular generalized gaussian random fields are studied certain uniqueness theorems are proved some qualitative properties of the set of the corresponding gibbs states are described

approximate solution to linear complex differential equation by a new approximate approach the purpose of this study is to give a taylor polynomial approximation for the solution of high order linear complex differential equations with variable coefficients and mixed conditions about any point for this purpose a matrix method is introduced this method is based on first taking the truncated taylor expansions of the functions in the complex differential equation and then substituting their matrix forms into the given equation hence the result matrix equation can be solved and the unknown taylor coefficients can be found approximately also examples that illustrate the pertinent features of the method are presented and the results are discussed

splitting in large dimension and infrared estimates ii moment inequalities this is the continuation of notes written for the nato asi conference in il ciocco september 96 consisting of the analysis of the links between estimating the splitting between the two first eigenvalues for the schrodinger operator h and the proof of infrared estimates for quantities attached to gaussian type measures these notes were mainly reporting on the old contributions of dyson frohlich glimm jaffe lieb simon and spencer in the 1970s in connection with more recent contributions of pastur khoruzhenko barbulyak and kondrat ev which treat in general more sophisticated models here we concentrate on the simplest model related to field theory and extend the results of barbulyak and kondrat ev by mixing ideas coming from pastur and khozurenko related to the use of bogolyubov s inequality with classical inequalities due to ginibre lebowitz sokal and others or in the case when the temperature t is zero by applying rather elementary estimates on schrodinger operators in order to find

approximate solution of complex differential equations for a rectangular domain with taylor collocation method in this study taylor collocation method is investigated for the approximate computation of high order linear complex differential equations using the collocation points on any rectangular domain in the complex plane the method transforms the given complex differential equation and the mixed conditions to matrix equation with unknown taylor coefficients by means of the obtained matrix equations the taylor coefficients can be easily computed hence the finite taylor series approach is obtained also examples are presented and the results are discussed

euclidean gibbs measures on loop lattices existence and a priori estimates we present a new method to prove existence and uniform a priori estimates for euclidean gibbs measures corresponding to quantum anharmonic crystals it is based first on the alternative characterization of gibbs measures in terms of their logarithmic derivatives through integration by parts formulas and second on the choice of appropriate lyapunov functionals

extensions and generalizations of a theorem of widder and of the theory of symmetric local semigroups the theory of symmetric local semigroups due to a klein and l landau j funct anal 44 1981 121 136 is generalized to semigroups indexed by subsets of rn for n 1 the result implies a similar result of a e nussbaum j funct anal 48 1982 213 223 it is further generalized to semigroups that are symmetric local in some directions and unitary in others the results are used to give a simple proof of a devinatz s duke math j 22 1955 185 192 and n i akhiezer s the classical moment problem and some related questions hafner new york 1965 generalization of a theorem of widder concerning the representation of functions as laplace integrals this result is extended to the representation as a laplace integral of a function taking values in b r the set of bounded linear operators on a hilbert space r also a theorem is proved encompassing both the result of devinatz and akhiezer and bochner s theorem on the representation of positive definite functions as fourier integrals

a taylor collocation method for the approximate solution of general linear fredholm volterra integro difference equations with mixed argument abstract in this paper a taylor collocation method is developed to find an approximate solution of high order linear fredholm volterra integro difference equations with variable coefficients under the mixed conditions the solution is obtained in terms of taylor polynomial near any point also examples are presented which illustrate the partinent features of the method and the results are discussed

the approximate solution of high order linear difference equations with variable coefficients in terms of taylor polynomials this paper presents a numerical method for the approximate solution of mth order linear difference equations with variable coefficients under the mixed conditions in terms of taylor polynomials about any point in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

ergodicity ofl2 semigroups and extremality of gibbs states we extend the classical results of holley stroock on the characterization of extreme gibbs states for the ising model in terms of the irreducibility resp ergodicity of the corresponding glauber dynamics to the case of lattice systems with unbounded linear spin spaces we first develop a general framework to discuss questions of this type using classical dirichlet forms on infinite dimensional state spaces and their associated diffusions we then describe concrete applications to lattice models with polynomial interactions i e the discretep d models of euclidean quantum field theory in addition we prove the equivalence of extremality and shift ergodicity for tempered gibbs states of these models and also discuss this question in the general framework

reflection positivity and monotonicity we prove general reflection positivity results on riemannian manifolds for both scalar fields and dirac fields and comment on applications to quantum field theory as another application we prove the inequality cd cn between dirichlet and neumann covariance operators on a manifold with a reflection

stationary parabolic anderson model and intermittency this paper is devoted to the analysis of the large time behavior of the solutions of the anderson parabolic problem frac partial u partial t kappa delta u xi x u when the potential x is a homogeneous ergodic random field on d our goal is to prove the asymptotic spatial intermittency of the solution and for this reason we analyze the large time properties of all the moments of the positive solutions this provides an extension to the continuous space d of the work done originally by gartner and molchanov in the case of the lattice d in the process of our moment analysis we show that it is possible to exhibit new asymptotic regimes by considering a special class of generalized gaussian fields interpolating continuously between the exponent 2 which is found in the case of bona fide continuous gaussian fields x and the exponent 3 2 appearing in the case of a one dimensional white noise finally we also determine the precise almost sure large time asymptotics of the positive solutions

multi parameter local semi groups of hermetian operators abstract it is proven that if a weakly continuous multi parameter family of hermetian operators in a hilbert space whose domains are not necessarily dense but whose union is dense and which has the semi group property for the parameters belonging to an open convex set in euclidean n space with 0 a limit point then the operators have self adjoint extensions which have a joint spectral representation

construction of a unique self adjoint generator for a symmetric local semigroup abstract a definition is given of a symmetric local semigroup of unbounded operators p t 0 t t for some t 0 on a hilbert space h such that p t is eventually densely defined as t 0 it is shown that there exists a unique unbounded below self adjoint operator h on h such that p t is a restriction of e th as an application it is proven that h0 v is essentially self adjoint where e th0 is an lp contractive semigroup and v is multiplication by a real measurable function such that v l2 e and e v l1 for some e 0

laguerre entire functions and the lee yang property laguerre entire functions are the polynomials of a single complex variable possessing real nonpositive zeros only or their limits on compact subsets of c these functions are employed to establish a property of isotropic i e o n invariant probability measures on r n email protected n it is called the lee yang property since in the case n 1 it corresponds to the property of the partition function of certain models of statistical physics first established by t d lee and c n yang a class of measures possessing this property is described certain connections of the lee yang property with other aspects of the analytic theory of probability measures are discussed

reconstruction of kubo martin schwinger structure from euclidean green functions the explicit form of the poisson kernel for the strip t z c 0 im z is used to derive some conditions for a state on a given c algebra dynamical system a t to be a kubo martin schwinger kms state reflection positivity and generalized periodicity of the euclidean green functions corresponding to are established and used for an analytic continuation to real time reflection positivity is also used for the construction of a cyclic and faithful representation of a t on some hilbert space a sufficient condition for the identification of the constructed dynamics with the modular group of a von neumann algebra generated by the representation is formulated in terms of the euclidean green functions

equivalence of euclidean and wightman field theories a new inversion formula for the laplace transformation of tempered distributions with supports in the closed positive semiaxis is obtained the inverse laplace transform of a tempered distribution is defined by means of a limit of a special distribution constructed from this distribution the weak spectral condition on the euclidean green s functions implies that some of the limits needed for the inversion formula exist for any euclidean green s function with an even number of variables we then prove that the initial osterwalder schrader axioms 1 and the weak spectral condition are equivalent with the wightman axioms

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

solving riccati differential equation using adomian s decomposition method in this paper we suggest a method to solve the matrix riccati differential equation the suggested method which we called multistage adomian s decomposition method madm can be considered as an extension of the adomian s decomposition method adm which is very efficient in solving a variety of differential and algebraic equations the solution is introduced in a recursive form which can be used to obtain the solution for the whole time horizon comparisons are made between madm and the exact solution and further between madm and different numerical methods

status of the differential transformation method abstract further to a recent controversy on whether the differential transformation method dtm for solving a differential equation is purely and solely the traditional taylor series method it is emphasized that the dtm is currently used often only as a technique for analytically calculating the power series of the solution in terms of the initial value parameters sometimes a piecewise analytic continuation process is implemented either in a numerical routine e g within a shooting method or in a semi analytical procedure e g to solve a boundary value problem emphasized also is the fact that at the time of its invention the currently used basic ingredients of the dtm that transform a differential equation into a difference equation of same order that is iteratively solvable were already known for a long time by the traditional taylor method users notably in the elaboration of software packages numerical routines for automatically solving ordinary differential equations at now the defenders of the dtm still ignore the though much better developed studies of the traditional taylor method users who in turn seem to ignore similarly the existence of the dtm the dtm has been given an apparent strong formalization set on the same footing as the fourier laplace or mellin transformations though often used trivially it is easily attainable and easily adaptable to different kinds of differentiation procedures that has made it very attractive hence applications to various problems of the taylor method and more generally of the power series method including noninteger powers has been sketched it seems that its potential has not been exploited as it could be after a discussion on the reasons of the misunderstandings which have caused the controversy the preceding topics are concretely illustrated it is concluded that for the sake of clarity when the dtm is applied to odes it should be mentioned that the dtm exactly coincides with the traditional taylor method contrary to what is currently written

continuous integral kernels for unbounded schrodinger semigroups and their spectral projections abstract by suitably extending a feynman kac formula of simon canad math soc conf proc 28 2000 317 we study one parameter semigroups generated by the negative of rather general schrodinger operators which may be unbounded from below and include a magnetic vector potential in particular a common domain of essential self adjointness for such a semigroup is specified moreover each member of the semigroup is proven to be a maximal carleman operator with a continuous integral kernel given by a brownian bridge expectation the results are used to show that the spectral projections of the generating schrodinger operator also act as carleman operators with continuous integral kernels applications to schrodinger operators with rather general random scalar potentials include a rigorous justification of an integral kernel representation of their integrated density of states a relation frequently used in the physics literature on disordered solids

analytic continuation of local representations of symmetric spaces we prove the following analytic continuation theorem which applies to any virtual representation of any symmetric space g k the problem of passing from the euclidean group to the poincare group appears first to have been addressed and solved this way by klein and landau let g be a lie group k a closed subgroup and an involutive automorphism with k as fixed point subgroup if g k m is the corresponding symmetric lie algebra we form g k im and let g denote the simply connected lie group with g as lie algebra we consider virtual representations of g on a fixed complex hilbert space h adopting the definitions due to j frohlich k osterwalder and e seiler in particular g 1 g possibly unbounded operators for g in a neighborhood of e in g we prove that every such continues analytically to a strongly continuous unitary representation of g on h our theorem extends results due to klein landau frohlich et al and others earlier for special cases previous results were known only for special g k and then only for certain

w kms structure from multi time euclidean green functions an axiomatic approach to the problem of reconstruction of a dynamics from the euclidean multi time green functions is proposed the existence of a w algebra on which the reconstructed dynamics acts as the canonical group of modular automorphisms is shown a condition under which the reconstruction from the one time and multi time green functions are equivalent is formulated

a priori estimates for symmetrizing measures and their applications to gibbs states abstract we prove existence and uniform a priori estimates for tempered gibbs states of certain classical lattice systems with unbounded spins nonharmonic pair potentials and infinite radius of interaction we use an alternative characterization of gibbs measures in terms of their radon nikodym derivatives w r t local shifts of the configuration space and corresponding integration by parts formulas

a numerical approximation based on the bessel functions of first kind for solutions of riccati type differential difference equations in this study a collocation method based on the bessel functions of first kind is given for the approximate solutions of the riccati differential difference equations under the mixed condition the method is presented with error analysis numerical examples are included to demonstrate the validity and applicability of the method and the comparisons are made with existing results

markov operators and os positive processes abstract we study factorization problems for markov operators between probability spaces the factorable operators are shown to be a weakly closed cone and examples of non factorable operators are constructed using c algebraic methods these ideas are applied to the study of os positive processes in discrete time and some examples of os positive processes are constructed which cannot be extended to markov processes

from the euclidean group to the poincare group via osterwalder schrader positivity given a continuous representation of the euclidean group inn 1 dimensions together with a covariant system of subspaces which satisfies osterwalder schrader positivity we construct a continuous unitary representation of the orthochronous poincare group inn 1 dimensions satisfying the spectral condition a similar result holds for the covering groups of the euclidean and poincare group

on the solution of the riccati equation by the taylor matrix method in this paper we suggest a matrix method to solve the riccati differential equation in terms of taylor polynomials this method is based on first taking the truncated taylor series of the function in equations and then substituting their matrix forms into the given equation hence the result matrix equation can be solved and the unknown taylor coefficients can be found approximately the solution is calculated in the form of a series with easily computable components the numerical results show the effectiveness of the method for this type of equation comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate

calculation of the taylor series expansion coefficients of the jacobian elliptic function introduction not much is known about the taylor series expansion coefficients of the jacobian elliptic functions sn x k cn x k and dn x k in handbooks only the first four or five terms are given see for instance abramowitz and stegun 1 p 575 hancock s book on elliptic functions 4 p 252 and p 486 or gradshteyn and ryshik 5 recently however alois schett gave a combinatorial expression of the coefficients of the taylor series expansion of sn x k and evaluated explicitly the first eight nontrivial terms see 6 p 146 his results were extended by dominique dumont 3 who gave a new combinatorial interpretation of the coefficients of sn x k it is often said that no recurrence formulae exist for these coefficients 1 p 575 or 6 p 143 what is meant by that is probably that no simple recurrence formulae exist using differential and other equations one can deduce a number of useful recurrence relations and with the help of the burmann lagrange theorem even a definite although hard to handle expression of the coefficients in question

translation invariant equilibrium states of ferromagnetic abelian lattice systems the structure of the set of all translation invariant equilibrium states is determined for all temperatures for which the free energy is differentiable models with several phase transitions are discussed rigorously

on virtual representations of symmetric spaces and their analytic continuation in this paper we study a problem in group representation theory motivated by relativistic quantum field theory one of the most powerful approaches to constructing models of relativistic quantum fields relies on euclidean field theory a description of quantum field theory in which time is purely imaginary the basic objects of euclidean field theory the euclidean green s or schwinger functions can often be expressed in terms of functional integrals the construction of quantum field models is thereby reduced to quadratures a well known simple example of this method is the reformulation of quantum mechanisms by means of wiener integrals the fundamental solution of the schrodinger equation analytically continued to imaginary time is given by the well known feynman kac formula see e g 15 19 in this approach one faces the problem of analytic continuation back to real time while this problem is elementary in non relativistic quantum mechanics it is rather intricate in relativistic quantum field theory for some class of relativistic quantum field theories a fairly general solution to this problem was given in papers by osterwalder and

stochastically positive structures on weyl algebras the case of quasi free states we consider quasi free stochastically positive ground and thermal states on weyl algebras in the imaginary time formulation in particular we obtain a new derivation of a general form of thermal quasi free state and give conditions when such a state is stochastically positive i e when it defines a periodic stochastic process with respect to imaginary time a so called thermal process then we show that the thermal process completely determines modular structure canonically associated with the quasi free thermal state on weyl algebra we discuss a variety of examples connected with free quantum field theories on globally hyperbolic stationary space times and models of quantum statistical mechanics

a new polynomial approach for solving difference and fredholm integro difference equations with mixed argument the purpose of this study is to give a taylor polynomial approximation for the solution of high order linear fredholm difference equations with variable coefficients and mixed argument under the mixed conditions about any point for this purpose a taylor matrix method is introduced this method is based on first taking the truncated taylor expansions of the functions in the difference equation and then substituting their matrix forms into the given equation hence the result matrix equation can be solved and the unknown taylor coefficients can be found approximately also examples that illustrate the pertinent features of the method are presented and the results are discussed

a quantum crystal with multidimensional anharmonic oscillators the quantum anharmonic crystal is made up of a large number of multidimensional anharmonic oscillators arranged in a periodic spatial lattice with a nearest neighbor coupling if the coupling coefficient is sufficiently small then there is a convergent expansion for the ground state of the crystal the estimates on the convergence are independent of the size of the crystal the proof uses the path integral representation of the ground state in terms of diffusion processes the convergence of the cluster expansion depends on the ergodicity properties of these processes

a taylor method for numerical solution of generalized pantograph equations with linear functional argument this paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument in this paper we introduce a numerical method based on the taylor polynomials for the approximate solution of the pantograph equation with retarded case or advanced case the method is illustrated by studying the initial value problems the results obtained are compared by the known results

quantum statistical mechanics for superstable interactions bose einstein statistics we study quantum systems of interacting bose particles confined to a bounded region ofr v for any superstable and strong lower regular interaction we obtain uniform bounds on the expectations of exponentials of local number operators for any activity and for any temperature the method we use here is an improvement over our previous method on the same subject as a consequence of the bounds any infinite volume limit states are entire analytic and locally normal furthermore under an integrability condition on the interaction the limit states are modular states in this case we use the green s function method to construct an infinite volume limit hilbert spece a strongly continuous time evolution group of unitary operators and an invariant vector moreover we prove the existence of the pressure and its independence of boundary conditions

thermal quantum fields with spatially cutoff interactions in 1 1 space time dimensions we construct interacting quantum fields in 1 1 space time dimensions representing charged or neutral scalar bosons at positive temperature and zero chemical potential our work is based on prior work by klein and landau and hoegh krohn generalized path space methods are used to add a spatially cut off interaction to the free system which is described in the araki woods representation it is shown that the interacting kms state is normal w r t the araki woods representation the observable algebra and the modular conjugation of the interacting system are shown to be identical to the ones of the free system and the interacting liouvillean is described in terms of the free liouvillean and the interaction

phase transitions on nonamenable graphs we survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph most attention is devoted to transitive graphs and trees

stochastic processes associated with kms states abstract a general mathematical framework is presented for the connection between quantum statistical mechanics and periodic stochastic processes stochastically positive kms systems periodic os positive stochastic processes periodic positive semigroup structures are defined and shown to be equivalent the objects of the tomita takesaki theory are explicitly exhibited in terms of the associated stochastic process and the statements of the theory are proven a two sided markov property for the stochastic process is related to a cyclicity property of the kms state perturbations are constructed by the feynman kac nelson formula the general framework is applied to kms states given by density matrices

on the phase diagram for a class of continuous systems some gibbsian aspects of the scheme known as the sine gordon transformation are discussed dobrushin lanford ruelle dlr equilibrium functional equations are introduced and analyzed certain uniqueness theorems are proved thermodynamical properties of the corresponding gibbs measures are discussed in particular the gibbs variational principle and the information gain principle are proved

uniqueness theorem for a class of continuous systems abstract we present a general abstract scheme for the analysis of kirkwood salsburg like identities the main conclusions obtained in this manner seem to be much better than those obtained by the standard contraction map principle approach detailed application of the general method to the theory of the classical continuous gas with a superstable interaction potential is given in particular it is proved that the whole resolvent set of the corresponding kirkwood salsburg operator corresponds to a region free of phase transitions of any order

a taylor collocation method for the numerical solution of complex differential equations with mixed conditions in elliptic domains an approximate method for solving higher order linear complex differential equations in the elliptic domains is proposed the approach is based on a taylor collocation method which consists of the matrix representation of expressions in the differential equation and the collocation points defined in the elliptic domain illustrative examples are included to demonstrate the validity and applicability of the technique and performed on the computer using a program written in maple9

a new collocation method for solution of mixed linear integro differential difference equations numerical solution of mixed linear integro differential difference equation is presented using chebyshev collocation method the aim of this article is to present an efficient numerical procedure for solving mixed linear integro differential difference equations our method depends mainly on a chebyshev expansion approach this method transforms mixed linear integro differential difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation the reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system maple10

algebraic solution of the lindblad equation for a collection of multilevel systems coupled to independent environments we consider the lindblad equation for a collection of multilevel systems coupled to independent environments the equation is symmetric under the exchange of the labels associated with each system and thus the open system dynamics takes place in the permutation symmetric subspace of the operator space the dimension of this space grows polynomially with the number of systems we construct a basis of this space and a set of superoperators whose action on this basis is easily specified for a given number of levels m these superoperators are written in terms of a bosonic realization of the generators of the lie algebra in some cases these results enable finding an analytic solution of the master equation using known lie algebraic methods to demonstrate this we obtain an analytic expression for the state operator of a collection of three level atoms coupled to independent radiation baths when analytic solutions are difficult to find the basis and the superoperators can be used to considerably reduce the computational resources required for simulations

a numerical approach for solving generalized abel type nonlinear differential equations in this paper a numerical power series algorithm which is based on the improved taylor matrix method is introduced for the approximate solution of abel type differential equations and also riccati differential equations the technique is defined and illustrated with some numerical examples the obtained results reveal that the method is very effective simple and valid high accuracy the method can be easily extended to other nonlinear equations

uniqueness of gibbs states for quantum lattice systems we prove uniqueness of euclidean gibbs states for certain quantum lattice systems with unbounded spins we use dobrushin s uniqueness criterion the necessary estimates for the vasershtein distance between the corresponding one point conditional distributions with boundary conditions differing only at one side are obtained by proving a log sobolev inequality on the infinite dimensional single spin loop spaces some important classes of concrete examples to which all this applies are discussed

compactness and the maximal gibbs state for random gibbs fields on a lattice we prove for a general class of gibbsian random field on that the set of tempered gibbs states is compact this class contains the euclidean random fields moreover if the interaction is attractive there is a unique minimal and maximal gibbs state and are unique translation invariant ant and have the global markov property we also prove that uniqueness of the tempered gibbs state is equivalent to the magnetizationsm qx being equal which is true if the pressure is differentiable

the double wedge algebra for quantum fields on schwarzschild and minkowski spacetimes we consider the klein gordon equation m 0 on the double schwarzschild wedge of the kruskal spacetime and construct the hartle hawking state h as a thermal state relative to the boulware quantization we prove that on the double wedge h is a pure state and in the corresponding representation the left and right wedgec algebras each have the reeh schlieder property while the corresponding von neumann algebras are typeiii1 factors which are dual to i e commutants of each other we discuss the extent to which these properties may generalize to non quasi free field theories

a new matrix method for solving two dimensional time dependent diffusion equations with dirichlet boundary conditions abstract this paper is devoted to develop a new matrix scheme for solving two dimensional time dependent diffusion equations with dirichlet boundary conditions we first transform these equations into equivalent integro partial differential equations pdes such these integro pdes contain both of the initial and boundary conditions and can be solved numerically in a more appropriate manner subsequently all the existing known and unknown functions in the latter equations are approximated by bernoulli polynomials and operational matrices of differentiation and integration together with the completeness of these polynomials can be used to reduce the integro pdes into the associated algebraic generalized sylvester equations for solving these algebraic equations an efficient krylov subspace iterative method i e bicgstab is implemented two numerical examples are given to demonstrate the efficiency accuracy and versatility of the proposed method

unitary representations of lie groups with reflection symmetry we consider the following class of unitary representations of some real lie groupgwhich has a matched pair of symmetries described as follows i supposeghas a period 2 automorphism and that the hilbert spaceh carries a unitary operatorjsuch thatj j i e selfsimilarity ii an added symmetry is implied ifh further contains a closed subspacek0having a certainorder covarianceproperty and satisfying thek0 restricted positivity v jv 0 v k0 where is the inner product inh from i ii we get an induced dual representation of an associated dual groupgc all three properties selfsimilarity order covariance and positivity are satisfied in a natural context whengis semisimple and hermitean but whengis the ax b group or the heisenberg group positivity is incompatible with the other two axioms for the infinite dimensional irreducible representations we describe a class ofg containing the latter two which admits a classification of the possible spacesk0 h satisfying the axioms of selfsimilarity and order covariance

path integral representation of fractional harmonic oscillator abstract fractional oscillator process can be obtained as the solution to the fractional langevin equation there exist two types of fractional oscillator processes based on the choice of fractional integro differential operators namely weyl and riemann liouville an operator identity for the fractional differential operators associated with the fractional oscillators is derived it allows the solution of fractional langevin equations to be obtained by simple inversion the relationship between these two fractional oscillator processes is studied the operator identity also plays an important role in the derivation of the path integral representation of the fractional oscillator processes relevant quantities such as two point and n point functions can be calculated from the generating functions

gentle perturbations of the free bose gas i it is demonstrated that the thermal structure of the noncritical free bose gas is completely described by certain periodic generalized gaussian stochastic process or equivalently by a certain periodic generalized gaussian random field elementary properties of this gaussian stochastic thermal structure are established gentle perturbations of several types of the free thermal stochastic structure are studied in particular new models of non gaussian thermal structures are constructed and a new functional integral representation of the corresponding euclidean time green functions is obtained rigorously

reflection positivity and conformal symmetry abstract a reflection positive hilbert space is a triple e e where e is a hilbert space a unitary involution and e a closed subspace on which the hermitian form v w v w is positive semidefinite for a triple g s where g is a lie group an involutive automorphism of g and s a subsemigroup invariant under the involution s s s 1 a unitary representation of g on e e is called reflection positive if g g and s e e this is the first in a series of papers in which we develop a new and systematic approach to reflection positive representations based on reflection positive distributions and reflection positive distribution vectors this approach is most natural to obtain classification results in particular in the abelian case among the tools we develop is a generalization of the bochner schwartz theorem to positive definite distributions on open convex cones we further illustrate our techniques with a non abelian example by constructing reflection positive distribution vectors for complementary series representations of the conformal group o 1 n 1 r of the sphere s n

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

dobrushin s uniqueness for quantum lattice systems with nonlocal interaction based on dobrushin s fundamental criterion we prove uniqueness of euclidean gibbs states for a certain class of quantum lattice systems with unbounded spins nonharmonic pair potentials and infinite radius of interaction the necessary estimates on dobrushin s coefficients are obtained from the log sobolev inequality which holds for the one point conditional distributions on the infinite dimensional single spin loop spaces

approximate solution of multi pantograph equation with variable coefficients this paper deals with the approximate solution of multi pantograph equation with nonhomogenous term in terms of taylor polynomials the technique we have used is based on a taylor matrix method in addition some numerical examples are presented to show the properties of the given method and the results are discussed

uniqueness and clustering properties of gibbs states for classical and quantum unbounded spin systems we consider quantum unbounded spin systems lattice boson systems in dimensional lattice space z under appropriate conditions on the interactions we prove that in a region of high temperatures the gibbs state is unique is translationally invariant and has clustering properties the main methods we use are the wiener integral representation the cluster expansions for zero boundary conditions and for general gibbs state and explicitly dependent probability estimates for one dimensional systems we show the uniqueness of gibbs states for any value of temperature by using the method of perturbed states we also consider classical unbounded spin systems we derive necessary estimates so that all of the results for the quantum systems hold for the classical systems by straightforward applications of the methods used in the quantum case

ergodicity for the stochastic dynamics of quasi invariant measures with applications to gibbs states the convex set maof quasi invariant measures on a locally convex spaceewith given shift radon nikodym derivatives i e cocycles a atk k k0 t ris analyzed the extreme points of maare characterized and proved to be non empty a specification of lattice type is constructed so that macoincides with the set of the corresponding gibbs states as a consequence via a well known method due to dynkin and follmer a unique representation of an arbitrary element in main terms of extreme ones is derived furthermore the corresponding classical dirichlet forms e d e and their associated semigroups t t t 0onl2 e are discussed under a mild positivity condition it is shown that mais extreme if and only if e d e is irreducible or equivalently t t t 0is ergodic this implies time ergodicity of associated diffusions applications to gibbs states of classical and quantum lattice models as well as those occuring in euclidean quantum field theory are presented in particular it is proved that the stochastic quantization of a guerra rosen simon gibbs state on d r2 ininfinite volumewith polynomial interaction is ergodic if the gibbs state is extreme i e is a pure phase which solves a long standing open problem

euclidean gibbs measures of interacting quantum anharmonic oscillators a rigorous description of the equilibrium thermodynamic properties of an infinite system of interacting dimensional quantum anharmonic oscillators is given the oscillators are indexed by the elements of a countable set mathbb l subset mathbb r d possibly irregular the anharmonic potentials vary from site to site and the interaction has infinite range the description is based on the representation of the gibbs states in terms of path measures the so called euclidean gibbs measures it is proven that a the set of such measures mathcal g rm t is non void and compact b every mu in mathcal g rm t obeys an exponential integrability estimate the same for the whole set mathcal g rm t c every mu in mathcal g rm t has a lebowitz presutti type support d mathcal g rm t is a singleton at high temperatures the case of attractive interaction and nu 1 is studied in more detail we prove that a mathcal g rm t 1 at low temperatures b mathcal g rm t 1 due to quantum effects and at a nonzero external field thereby a qualitative theory of phase transitions and quantum effects which interprets most important experimental data known for the corresponding physical objects is developed

small mass implies uniqueness of gibbs states of a quantum crystal a model of interacting quantum particles performing one dimensional anharmonic oscillations around their equilibrium positions which form a lattice d is considered for this model it is proved that the set of tempered euclidean gibbs measures is a singleton provided the particle mass is less than a certain bound m which is independent of the temperature 1 this settles a problem that was open for a long time and is an essential improvement of a similar result proved before by the same authors 5 where the bound m depended on in such a way that m ast beta rightarrow 0 as beta rightarrow infty

traces of harmonic functions and a new path space for the free quantum field the infinite dimensional ornstein uhlenbeck process associated with the free field of quantum field theory on rd is reconstructed as the traces of random harmonic functions these harmonic functions are the solutions of a generalised dirichlet problem with a random schwartz distribution as boundary data a detailed analysis of the associated trace space which is a subspace of a scaled sobolev space of negative fractional order leads to a new path space for the free field as well as to a convenient regularisation method because of the particular form of the trace space it is possible to give an explicit formula for the transition function of the process furthermore the dependence of the path space on the dimension d is discussed and a formulation of our result in terms of abstract wiener spaces is given

an efficient approach for solving the riccati equation with fractional orders the present study introduces a novel and simple analytical method for the solution of fractional order riccati differential equation in this approach the solution considered as a taylor series expansion converges rapidly to the nonlinear problem new homotopy perturbation method nhpm depends only on two components of the homotopy series the method is illustrated by applications and the results obtained are compared with those of the exact solution moreover comparing the methodology with some known techniques shows that the present approach is relatively easy and efficient

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

low temperature expansions for the gibbs states of quantum ising lattice systems by conditioning on a subalgebra of observables we develop low temperature expansions for certain quantum ising models which include transverse ising models and anisotropic xy models these expansions are majorized by the series for functions solving certain nonlinear partial differential equations convergence for the majorizing series follows from cauchy kowalevski theory applied to the partial differential equations the low temperature expansions imply long range order

phase transitions and algebra of fluctuation operators in an exactly soluble model of a quantum anharmonic crystal a complete description of the fluctuation operator algebra is given for a quantum crystal showing displacement structural phase transitions in the one phase region the fluctuations are normal and its algebra is non abelian in the two phase region and on the critical line t c 0 the momentum fluctuation is normal the displacement is critical and the algebra is abelian att c 0 quantum phase transition this algebra is non abelian with abnormal displacement and supernormal squeezed momentum fluctuation operators both being dimension dependent