on the relaxation properties of stochastic ginzburg landau type models abstract considering the project of studying the basic properties of commonly used stochastic nonlinear dynamical systems we investigate and present new properties of the relaxation to equilibrium of stochastic ginzburg landau type models with polynomial interaction with nonlocal nonlinear terms in our approach we map the initial stochastic model on a imaginary time quantum field theory we show that with a suitable nonlocality a two particle bound state appears below the two particle threshold even if all the coefficients in the polynomial interaction are positive the existence of such a bound state changes the relaxation rate to equilibrium and so it is an experimentally observable effect of physical interest moreover we show that the bound state masses are sensitive to changes in the rate of nonlocality which may favor other phenomena related to the low lying spectrum of the dynamics generator our results involve a perturbative analysis supported by previous rigorous results in the computation of the bound state mass we use a bethe salpeter equation in the ladder approximation

boundedness of inhomogeneous journe s type operators on multi parameter local hardy spaces abstract in this paper we establish a boundedness criterion of a class of inhomogeneous journe s type of multi parameter singular integral operators on multi parameter local hardy spaces h p r n 1 r n k recently introduced in ding et al 2019 the consideration of such inhomogeneous journe s type of multi parameter singular integral operators is motivated by the study of the multi parameter pseudo differential operators on multi parameter local hardy spaces the lack of the vanishing moments of both the corresponding functions in the multi parameter local calderon reproducing formula and the multi parameter local littlewood paley stein functions as well as the lack of vanishing moments of atoms in the multi parameter local hardy spaces h p r n 1 r n k with large size all create substantial difficulties in our proofs

a liouville type theorem for schrodinger operators in this paper we prove a sufficient condition in terms of the behavior of a ground state of a symmetric critical operator p 1 such that a nonzero subsolution of a symmetric nonnegative operator p 0 is a ground state particularly if p j v j for j 0 1 are two nonnegative schrodinger operators defined on omega subseteq mathbb r d such that p 1 is critical in with a ground state the function psi nleq 0 is a subsolution of the equation p 0 u 0 in and satisfies psi leq c varphi in then p 0 is critical in and psi is its ground state in particular psi is up to a multiplicative constant the unique positive supersolution of the equation p 0 u 0 in similar results hold for general symmetric operators and also on riemannian manifolds

on the toda and kac van moerbeke systems given a solution of the toda lattice we explicitly construct a solution of the kac van moerbeke system related to each other by a miura type transformation as an illustration of our method we derive the n soliton solutions of the kac van moerbeke lattice

calderon zygmund operators on product hardy spaces let t be a product calderon zygmund singular integral introduced by journe using an elegant rectangle atomic decomposition of hp rn rm and journe s geometric covering lemma r fefferman proved the remarkable hp rn rm lp rn rm boundedness of t in this paper we apply vector valued singular integral calderon s identity littlewood paley theory and the almost orthogonality together with fefferman s rectangle atomic decomposition and journe s covering lemma to show that t is bounded on product hp rn rm for max nn e mm e p 1 if and only if t1 1 t2 1 0 where e is the regularity exponent of the kernel of t

approach to equilibrium of particles diffusing on curved surfaces abstract we present a simple numerical analysis of the diffusion on a curved surface given by the equation r 0 in a finite domain d r 3 the first non vanishing eigenvalue of the beltrami laplace operator with the reflecting boundary conditions is determined in our simulations for the p d g s s1 and i wp nodal periodic surfaces where d is their respective cubic unit cell we observe that the first eigenvalue for the surfaces of simple topology p d g i wp is smaller than for the surfaces of complex topology s s1

multi parameter singular radon transforms iii real analytic surfaces abstract the goal of this paper is to study operators of the form t f x x f t x k t d t where is a real analytic function defined on a neighborhood of the origin in t x r n r n satisfying 0 x x is a cutoff function supported near 0 r n and k is a multi parameter singular kernel supported near 0 r n a main example is when k is a product kernel we also study maximal operators of the form m f x x sup 0 1 n 1 t 1 f 1 t 1 n t n x d t we show that m is bounded on l p 1 p we give conditions on under which t is bounded on l p 1 p these conditions hold automatically when k is a calderon zygmund kernel this is the final paper in a three part series the first two papers consider the more general case when is c

on the existence of the n body efimov effect abstract in this paper we prove the existence of the efimov effect for n body quantum systems with n 4 under the conditions that the bottom of the essential spectrum e 0 of the n body operator is attained by the spectra of a unique three cluster subhamiltonian and its three associated two cluster subhamiltonians and that at least two of these two cluster subhamiltonians have a resonance at the threshold e 0 we give a lower bound of the form c 0 log e 0 for the number of eigenvalues on the left of 0 where c 0 is a positive constant depending only on the reduced masses in the three cluster decomposition we also obtain a lower bound on the number of discrete eigenvalues in coupling constant perturbation

one particle subspaces in the stochastic xy model we study the spectrum of the generatorh of the glauber dynamics for a model of planar rotators on a lattice in the case of a high temperature 1 we construct two so called one particle subspacesh forh and describe the spectrum of the generator in these subspaces as a consequence we find time asymptotics of the correlations for the glauber dynamics

dynamical systems on infinitely sheeted riemann surfaces this paper is part of a program that aims to understand the connection between the emergence of chaotic behaviour in dynamical systems in relation with the multi valuedness of the solutions as functions of complex time in this work we consider a family of systems whose solutions can be expressed as the inversion of a single hyperelliptic integral the associated riemann surface r c is known to be an infinitely sheeted covering of the complex time plane ramified at an infinite set of points whose projection in the plane is dense the main novelty of this paper is that the geometrical structure of these infinitely sheeted riemann surfaces is described in great detail which allows to study global properties of the flow such as asymptotic behaviour of the solutions periodic orbits and their stability or sensitive dependence on initial conditions the results are then compared with a numerical integration of the equations of motion following the recent approach of calogero the real time trajectories of the system are given by paths on r that are projected to a circle on the complex plane due to the branching of r the solutions may have different periods or may not be periodic at all

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

hormander type theorems for multi linear and multi parameter fourier multiplier operators with limited smoothness abstract the main purpose of this paper is three fold first of all we are concerned with the limited smoothness conditions in the spirit of hormander on the multi linear and multi parameter coifman meyer type fourier multipliers studied by c muscalu j pipher t tao c thiele 2004 2006 where they established the l r estimates for the multiplier operators under the assumption that the multiplier has smoothness of sufficiently large order under our limited smoothness assumption we will prove the l p 1 l p n l r boundedness with 1 p 1 1 p n 1 r for 1 p 1 p n and 0 r second our proof of l r estimates also offers a different and more direct approach than the one given in muscalu et al 2004 2006 where they use the deep analysis of multi linear and multi parameter paraproducts third we also prove a hormander type multiplier theorem in the weighted lebesgue spaces for such operators when the fourier multiplier is only assumed with limited smoothness

schrodinger operators with few bound states we show that whole line schrodinger operators with finitely many bound states have no embedded singular spectrum in contradistinction we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast we also prove the following result for one and two dimensional schrodinger operators h with bounded positive ground states given a potential v if both h v are bounded from below by the ground state energy of h then v 0

subcriticality and gaugeability of the schrodinger operator we investigate a schrodinger operator 2 v in r d d 3 with a potential v in the class k d satisfying a similar kato condition at infinity and prove an equivalence theorem connecting various conditions on subcriticality strong positivity and gaugeability of the operator

lp norms of non critical schrodinger semigroups we consider schrodinger semigroups e th h v on rn with v c x 2 as x 0 c 12 n 2 2 with h 0 we determine the exact power law divergence of e th p p and of some e th q p as maps from lp to lq the results are expressed most naturally in terms of the power for which there exists a positive resonance such that h 0 x x

multi parameter singular radon transforms ii the lp theory the purpose of this paper is to study the l2 boundedness of operators of the form f x f t x k t dt where t x is a c function defined on a neighborhood of the origin in t x n n satisfying 0 x x is a c cut off function supported on a small neighborhood of 0 n and k is a multi parameter singular kernel supported on a small neighborhood of 0 n the goal is given an appropriate class of kernels k to give conditions on such that every operator of the above form is bounded on l2 the case when k is a calderon zygmund kernel was studied by christ nagel stein and wainger we generalize their conditions to the case when k has a multi parameter structure for example when k is given by a product kernel even when k is a calderon zygmund kernel our methods yield some new results this is the first paper in a three part series the later two of which are joint with e m stein the second paper deals with the related question of lp boundedness while the third paper deals with the special case when is real analytic

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

multiparameter operators and sharp weighted inequalities this article is concerned with the operators from harmonic analysis which are naturally associated to a multiple parameter family of dilations we are especially interested here in dealing with questions from the theory of such operators whose answers cannot be obtained by a reduction to the case of product operators we also introduce a new tool in order to carry out this multiparameter analysis which comes from the classical theory this is the concept of sharp weighted inequalities where for operators such as the hilbert transform or classical square function one asks for hilbert space inequalities which imply the correct estimates of the norm on lp of these operators as p becomes large 0 introduction in this article our main goal is to study classes of oper ators associated with multiparameter groups of dilations on rn in some sense the consideration of these operators is a natural next step or simplest case after those of the classical calderon zygmund theory and the product space theory it turns out that in order to obtain estimates for the multiparameter operators we shall be led to the following type of extremely classical question how does one obtain l2 estimates on r1 for the hilbert transform or other singular integrals which yield sharp estimates on the lp rl operator norm of h as p approaches 1 or oc these sharp weighted inequalities are of some independent interest and their study will be seen to be essentially equivalent to the deep work of chang wilson and wolff 3 on the exponential square integrability of functions whose area integral is bounded they also have other applications for example they provide an immediate extension of the chang wilson wolff estimates to product spaces as given by pipher in 12 at this point let us be more specific about the exact nature of the multipa rameter operators which we shall consider consider in r3 the dilations given by ps t x y z sx ty9 stz for s91 0 then there is a maximal operator first considered by zygmund ml and singular integrals tl introduced by ricci and stein 13 naturally associated to these dilations the maximal operator af3 is defined by

a class of integrable flows on the space of symmetric matrices for a given skew symmetric real n n matrix n the bracket x y n xny ynx defines a lie algebra structure on the space sym n n of symmetric n n real matrices and hence a corresponding lie poisson structure the purpose of this paper is to investigate the geometry integrability and linearizability of the hamiltonian system dot x x 2 n or equivalently in lax form the equation dot x x xn nx on this space along with a detailed study of the poisson geometry itself if n has distinct eigenvalues it is proved that this system is integrable on a generic symplectic leaf of the lie poisson structure of sym n n this is established by finding another compatible poisson structure

uniqueness of the stochastic dynamics for continuous spin systems on a lattice abstract we prove the uniqueness of the stochastic dynamics associated with gibbs measures on infinite products of compact riemannian manifolds

lp norms of nonnegative schrodinger heat semigroup and the large time behavior of hot spots this paper is concerned with the cauchy problem for the heat equation with a potential p tu u v x uin rn 0 u x 0 x in rn where t t n 3 l2 rn and v v x is a smooth nonpositive and radially symmetric function having quadratic decay at the space infinity in this paper we assume that the schrodinger operator h v is nonnegative on l2 rn and give the exact power decay rates of lq norm q 2 of the solution e th of p as t furthermore we study the large time behavior of the solution of p and its hot spots

positive solutions and large time behaviors of schrodinger semigroups simon s problem let h v where v is a multiplication operator by a real valued function v x on rn which is uniformly holder continuous and 1 x 2 2 v x l rn for some 4 the relationship between existence of positive solutions with growth conditions of hg 0 and asymptotic behaviors as t of e th is established using it b simon s problem for h on r2 is solved

large time behavior of the heat kernel and the behavior of the green function near criticality for nonsymmetric elliptic operators abstract let p be a second order elliptic operator which is defined on a domain r n and let w x be a nonzero nonnegative function assume that p p w x admits a positive minimal green function g p x y for all lim 0 g p x y f x y where f 0 for x y unless p and p admit ground states x and x respectively with eigenvalue zero such that z w z z dz in this case f x y x y z w z z dx moreover if z z dz then the heat kernel k x y t of the parabolic equation lu u t pu 0 satisfies lim t k x y t x y z z dz

constructing solutions of the mkdv equation abstract using commutation methods i e n 1 supersymmetry underlying miura s transformation an explicit construction of solutions of the modified korteweg de vries equation given a solution of the ordinary korteweg de vries equation is provided

asymptotic expansions in time for solutions of schrodinger type equations let ia i p d v be a dissipative operator in l2 rn where p d is an elliptic differential operator of order m with real constant coefficients and v is a compact operator from the weighted sobolev space hm s rn to h0 p s rn s r for some m 1 let r z be the resolvent of a then an asymptotic expansion of r z as z approaches a critical value of the polynomial p is given the coefficient operators in the expansion are computed explicitly by using the resolvent expansion and the results of m murata 9 an asymptotic expansion of e ita as t is given

harmonic analysis on product spaces rn x r and commute with the dilations p33 8 x2 x2 61x1 62x2 for x1 e rn x2 e rm and 61 82 0 then the corresponding operators are related to many problems in the theory of multiple fourier integrals see zygmund 21 the maximal operator which commutes with the p31382 is the jessen marcinkiewicz and zygmund 14 strong maximal operator ms defined by 1 ms f x1 x2 sup ri a y1 y2 1 dy1 dy2 xl x2 gr jr1

boundedness criterion of journeʼs class of singular integrals on multiparameter hardy spaces abstract this article concerns nonconvolutional type operators also known as journe s type operators associated with a multiparameter family of dilations given by x 1 x 2 x m 1 x 1 2 x 2 m x m where x 1 r n 1 x 2 r n 2 x m r n m and m 3 we are especially interested in the boundedness of such operators on the multiparameter hardy spaces this work is motivated by pipher s result on the boundedness of these operators from the multiparameter h p spaces to l p spaces for 0 p 1 and journe s counter example which shows that the number of parameter plays a crucial role in boundedness of singular integral operators on multiparameter hardy spaces journe s work shows that there is a sharp difference between the situations for two and three or more parameters we establish in this paper the necessary and sufficient conditions under which the singular integral operators in journe s class are bounded on multiparameter h p spaces 0 p 1 with arbitrary number of parameters

hyperelliptic prym varieties and integrable systems we introduce two algebraic completely integrable analogues of the mumford systems which we call hyperelliptic prym systems because every hyperelliptic prym variety appears as a fiber of their momentum map as an application we show that the general fiber of the momentum map of the periodic volterra lattice is an affine part of a hyperelliptic prym variety obtained by removing n translates of the theta divisor and we conclude that this integrable system is algebraic completely integrable

ornstein zernike asymptotics for a general two particle lattice operator we study the asymptotic behavior of correlations for a general two particle operator mathcal t acting on the hilbert space ell 2 mathbb z d times mathbb z d for all dimension d 1 2 mathcal t is written as the sum of a main term and a small interacting term a form which appears in many problems if the interacting term is small we give a complete description of the asymptotics for large t of the correlations mathcal t t f 1 f 2 t 1 2 ldots for f 1 f 2 in some suitable class the asymptotics is of the ornstein zernike type i e exponential with a power law factor which is t d for d 3 but for d 1 2 it can be anomalous and is determined by the interacting term

relative entropy and mixing properties of infinite dimensional diffusions let be a diffusion process taking values on the infinite dimensional space t z where t is the circle and with components satisfying the equations d i i dw i b i dt for some coefficients i and b i i z suppose we have an initial distribution and a sequence of times t n such that lim n s tn exists where s t is the semi group of the process we prove that if i and b i are bounded of finite range have uniformly bounded second order partial derivatives and inf i i 0 then is invariant

singular integrals with flag kernels and analysis on quadratic cr manifolds abstract we study a class of operators on nilpotent lie groups g given by convolution with flag kernels these are special kinds of product type distributions whose singularities are supported on an increasing subspace 0 v1 vk g we show that product kernels can be written as finite sums of flag kernels that flag kernels can be characterized in terms of their fourier transforms and that flag kernels have good regularity restriction and composition properties we then apply this theory to the study of the b complex on certain quadratic cr submanifolds of c n we obtain lp regularity for certain derivatives of the relative fundamental solution of b and for the corresponding szego projections onto the null space of b by showing that the distribution kernels of these operators are finite sums of flag kernels

asymptotics of eigenvalues of two particle schrodinger operators on lattices with zero range interaction the hamiltonian of a system of two quantum mechanical particles moving on the d dimensional lattice z d and interacting via zero range attractive pair potentials is considered for the two particle energy operator h k k t d d thetwo particlequasi momentum theexistenceofaunique positive eigenvalue z k above the upper edge of the essential spectrum of h k is proven and asymptotics for z k are found when approaches to some 0 k and k 0

weighted l p estimates for rough bi parameter fourier integral operators abstract fourier integral operators play an important role in applications to partial differential equations the main purpose of this paper is to investigate the weighted l p boundedness of the bi parameter fourier integral operators or fios for short of the following form t f x r n 1 r n 2 e i x a x f d d where for x x 1 x 2 r n 1 r n 2 r n 1 0 and r n 2 0 the amplitude function a x l b s m and the phase function is of the form x 1 x 1 2 x 2 when 1 l 2 r n 1 r n 1 0 2 l 2 r n 2 r n 2 0 satisfy certain differential conditions we will establish the weighted l w p boundedness of t under the assumption that the weight w is in the class of the product muckenhoupt a p r n 1 r n 2 weights

comportement des noyaux de la chaleur des operateurs de schrodinger et applications a certaines equations paraboliques semi lineaires abstract we consider general schrodinger operators on domains of riemannian manifolds with possibly exponential volume growth we prove sharp large time gaussian upper bounds these bounds are then used to prove new l p l p estimates for the corresponding semigroups applications to semi linear parabolic equations are given

criticality theory for schrodinger operators with singular potential abstract this paper extends a weighted spectral gap classification for schrodinger operators v to a large class of locally integrable potentials v called balanced potentials here harnack s inequality will not hold in general for such potentials in addition to the standard potentials in the space l l o c p for some p n 2 this class contains potentials for which v is locally bounded with no restriction on v or more generally those for which the positive and negative singularities are separated as well as those for which the interaction of the positive and negative singularities is not too strong we also establish for such potentials under the additional assumption that v l l o c p for some p n 2 versions of the agmon allegretto piepenbrink principle characterising the subcritical critical operators in terms of the cone of positive distributional super solutions

linear r matrix algebra for classical separable systems we consider a hierarchy of the natural type hamiltonian systems of n degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates we give a lax representation in terms of 2 2 matrices for the whole hierarchy and construct the associated linear r matrix algebra with the r matrix dependent on the dynamical variables a yang baxter equation of dynamical type is proposed using the method of variable separation we provide the integration of the systems in classical mechanics constructing the separation equations and hence the explicit form of action variables the quantization problem is discussed with the help of the separation variables

the threshold effects for the two particle hamiltonians on lattices for a wide class of two body energy operators h k on the d dimensional lattice d d 3 k being the two particle quasi momentum we prove that if the following two assumptions i and ii are satisfied then for all nontrivial values k k 0 the discrete spectrum of h k below its threshold is non empty the assumptions are i the two particle hamiltonian h 0 corresponding to the zero value of the quasi momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and ii the one particle free hamiltonians in the coordinate representation generate positivity preserving semi groups

the efimov effect of three body schrodinger operators abstract the author studies the efimov effect of three body schrodinger operators by use of the variational method the effect is proved under the assumptions that pair potentials fall off with order o x 2 at infinity and that all three two body subsystems have a zero resonance energy

on the connection between schrodinger and dirichlet forms relations between schrodinger forms associated with schrodinger operators in l2 dnx rn open n 1 and the corresponding dirichlet forms are investigated various concrete examples are presented

on the spectrum of an hamiltonian in fock space discrete spectrum asymptotics a model operator h associated with the energy operator of a system describing three particles in interaction without conservation of the number of particles is considered the location of the essential spectrum of h is described the existence of infinitely many eigenvalues resp the finiteness of eigenvalues below the bottom ess h of the essential spectrum of h is proved for the case where the associated friedrichs model has a threshold energy resonance resp a threshold eigenvalue for the number n z of eigenvalues of h lying below z ess h the following asymptotics is found lim limits z to tau rm ess h 0 frac n z log z tau rm ess h u 0 0 u 0 infty

nonlocal interactions and the excitation spectrum in lattice quantum scalar field models we determine the effects of nonlocal nonlinear interactions on the excitation spectrum of lattice quantum field scalar models we consider perturbations of a quantized discrete string formally self adjoint hamiltonian operator on the lattice d and with a large mass coefficient for the quadratic term the low lying energy momentum spectrum has an isolated dispersion curve and a two particle first band we analyse a ladder approximation of the bethe salpeter equation on the lattice for a weak perturbation of the type d 6 6 v 6 0 and consider the spectral interval starting at zero and extending to near the three particle threshold for space dimension d 1 2 and v 1 4 we find that a bound state occurs either below if 1 0 but not both this agrees with recent results where bound states were obtained for the stochastic dynamics generator associated with the relaxation rate to equilibrium in weakly coupled stochastic ginzburg landau models with continuous time and on a spatial lattice d these results are in contrast however with those obtained for v 2 3 for this case surprisingly we show that stable particles exist simultaneously above and below the band for d 1 2 regardless of the sign of the coupling 2 if v 3 2 2 the ladder analysis is inconclusive if d 3 4 no bound states exist in the spectral region we consider

multi hamiltonian structures for a class of degenerate completely integrable systems in this paper a class of degenerate i e associated to a degenerate poisson structure completely integrable systems is studied which generalizes the so called odd and even master systems introduced and studied by mumford and by vanhaecke it is shown that all these completely integrable systems called the generalized master systems admit a multi hamiltonian formulation and a systematic construction of this multi hamiltonian structure is described

spectral analysis of a stochastic ising model in continuum we consider an equilibrium stochastic dynamics of spatial spin systems in r d involving both a birth and death dynamics and a spin flip dynamics as well using a gen eral approach to the spectral analysis of corresponding markov generator we estimate the spectral gap and construct one particle invariant subspaces for the generator

and variants of multiparameter calderon zygmund theory we consider calderon zygmund operators on product domains under certain weak conditions on the kernel a singular integral operator can be proved to be bounded on h p r r r 0 p 1 if its behaviour on l 2 and on certain scalar valued and vector valued rectangle atoms is known another result concerns an extension of the authors results on l p variants of calderon zygmund theory 1 23 to the product domain setting as an application one obtains estimates for fourier multipliers and pseudo differential operators

spectral analysis of weakly coupled stochastic lattice ginzburg landau models we consider the relaxation to equilibrium of solutions t 0 of stochastic dynamical langevin equations with white noise and weakly coupled ginzburg landau interactions using a feynman kac formula which relates stochastic expectations to correlation functions of a spatially non local imaginary time quantum field theory we obtain results on the joint spectrum of h where h is the self adjoint positive generator of the semi group associated with the dynamics and p j j 1 d are the self adjoint generators of the group of lattice spatial translations we show that the low lying energy momentum spectrum consists of an isolated one particle dispersion curve and for the mass spectrum energy momentum at zero momentum besides this isolated one particle mass we show using a bethe salpeter equation the existence of an isolated two particle bound state if the coefficient of the quartic term in the polynomial of the ginzburg landau interaction is negative and d 1 2 otherwise there is no two particle bound state asymptotic values for the masses are obtained

anderson localization for time quasi periodic random schrodinger and wave equations we prove that at large disorder with large probability and for a corresponding set of diophantine frequencies of large measure anderson localization in d is stable under localized time quasi periodic perturbations by proving that the associated quasi energy operator has pure point spectrum the main tools are the frohlich spencer mechanism for the random component and the bourgain goldstein schlag mechanism for the quasi periodic component this paper paves the way for the construction of time quasi periodic kam type of solutions of non linear random schrodinger equations in bw

dirichlet parabolicity and l1 liouville property under localized geometric conditions abstract we shed a new light on the l 1 liouville property for positive superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space we also present examples in any dimension showing that the l 1 liouville property is strictly weaker than the stochastic completeness of the manifold the main tool in our investigations is represented by the potential theory of a manifold with boundary subject to dirichlet boundary conditions the paper incorporates under a unifying viewpoint some old and new aspects of the theory with a special emphasis on global maximum principles and on the role of the dirichlet green s kernel

large time behavior of the lp norm of schrodinger semigroups abstract letv 0 v c0 rv with v 3 be such that h 1 2 v 0 but for any e 0 1 2 1 e v is not positive we determine the exact rate of divergence of the norm of e th as a map from l to l a number of related problems are discussed

energy momentum spectrum of some two particle lattice schrodinger hamiltonians we determine the excitation spectrum of some one and two particle z d lattice schrodinger hamiltonians they occur as approximate hamiltonians for the low lying energy momentum spectrum of diverse infinite lattice nonlinear quantum systems a unitary staggering transformation relates the low energy momentum spectrum to the high energy momentum spectrum of the transformed operators a feature for the one particle repulsive delta function hamiltonian is that in addition to the continuous band spectrum there is a bound state above the band and the repulsive case spectrum and scattering can be obtained from the attractive potential case by staggering for the two particle pair potential hamiltonian there are commuting self adjoint energy momentum operators and we determine the joint spectrum for the case of a lambda delta pair potential and equal particle masses for arbitrarily small lambda lambda or 3 there is no bound state for small system momentum but a bound state exists below the band if the momentum is large we find that the binding energy is an increasing function of the system momentum the existence of this bound state is in contrast with the continuum case where the birman schwinger bound excludes negative energy bound states for small couplings this bound state is absent if the two masses are different other spectral results are also obtained for the large coupling case an eigenfunction expansion that uses products of plane waves in the sum and difference coordinates is used to obtain the spectral results

stochastic models of interacting systems interacting particle systems is by now a mature area of probability theory but one that is still very active we begin this paper by explaining how models from this area arise in fields such as physics and biology we turn then to a discussion of both older and more recent results about them concentrating on contact processes voter models and exclusion processes these processes are among the most studied in the field and have the virtue of relative simplicity in their description which permits us to address the fundamental issues about their behavior without dealing with the extra complications that models from specific areas of application would require

global bounds of schrodinger heat kernels with negative potentials abstract we obtain global in time bounds for the heat kernel g of the schrodinger operator l v the potential v satisfies v x c d x b near infinity with b 0 the result can be described as follows suppose l is positive and b 2 then g g x t y 0 has a global upper bound which is a standard gaussian times a polynomial function of x y t however when l is not nonnegative and negative eigenvalues exist g g x t 0 0 is bounded below by a standard gaussian times ectwithc 0 in other words the growth rate of g x t 0 0 is comparable with the heat kernel of c regardless how fast v decays near infinity

change of stability for schrodinger semigroups when the parameter x varies in the positive real axis here m is some bounded and continuous weight function being positive somewhere and u0 an initial condition the stability is understood as stability in the norm we point out that by duality and interpolation we also get the stability in lp r 1 g p oo the question of stability of the zero solution is closely related to the existence of a principal eigenvalue for the elliptic eigenvalue problem

the threshold effects for a family of friedrichs models under rank one perturbations a family of friedrichs models under rank one perturbations h p p 3 0 associated to a system of two particles on the three dimensional lattice z3 is considered we prove the existence of a unique eigenvalue below the bottom of the essential spectrum of h p for all non trivial values of p under the assumption that h 0 has either a threshold energy resonance virtual level or a threshold eigenvalue the threshold energy expansion for the fredholm determinant associated to a family of friedrichs models is also obtained

interband spectrum of weakly coupled stochastic lattice ginzburg landau models we analyze the excitation spectrum of the generator associated with the relaxation rate to equilibrium in weakly coupled stochastic ginzburg landau models on a spatial lattice z d the spectrum has a quasiparticle interpretation depending on d and on the specific interaction by solving the bethe salpeter equation in the ladder approximation we show the existence of a stable particle above the upper envelope of the two particle band possessing a concave dispersion curve this result furthers our knowledge about the spectrum of the stochastic dynamics generator

on the spectrum of the generator of an infinite system of interacting diffusions we study the spectrum of the operator generating an infinite dimensional diffusion process t in space here is a natural t invariant measure on which is a gibbs distribution corresponding to a formal hamiltonian h of an anharmonic crystal with a value of the inverse temperature 0 for small enough we establish the existence of an l invariant subspace such that has a distinctive character related to a quasi particle picture in particular has a lebesgue spectrum separated from the rest of the spectrum of l and concentrated near a point 1 0 giving the smallest non zero eigenvalue of a limiting problem associated with 0

sharp condition on the decay of the potential for the absence of a zero energy ground state of the schrodinger equation the authors prove a sharp criterion on the decay of the potential of a schrodinger operator on r3 that ensures the absence of a zero energy ground state this condition complements results due to simon and lieb 1981 they also give an improved version of their results which however is still not optimal

on the number of eigenvalues of a model operator related to a system of three particles on lattices a model operator h associated to a system of three particles on the threedimensional lattice 3 that interact via nonlocal pair potentials is studied the following results are established i the operator h has infinitely many eigenvalues lying below the bottom of the essential spectrum and accumulating at this point if both the friedrichs model operators h mu alpha 0 1 2 have threshold resonances ii the operator h has finitely many eigenvalues lying outside the essential spectrum if at least one of the operators h mu alpha 0 1 2 has a threshold eigenvalue

puiseux series expansion for an eigenvalue of the generalized friedrichs model with perturbation of rank 1 a family h p 0 of the generalized friedrichs model with perturbation of rank 1 associated with a system of two particles moving on the one dimensional lattice is considered the existence of a unique eigenvalue e p of the operator h p lying below the essential spectrum is proved for any p from a neighborhood of the origin the puiseux series expansion for eigenvalue e p at the point p 0 is found moreover the asymptotics for e p is established as

the log sobolev inequality for weakly coupled lattice fields we consider a ferromagnetic spin system with unbounded interactions on the d dimensional integer lattice d 1 under mild assumptions on the one body interactions so that arbitrarily deep double wells are allowed we prove that if the coupling constants are small enough then the finite volume gibbs states satisfy the log sobolev inequality uniformly in the volume and the boundary condition

structure of positive solutions to δ v u 0 in rn on etudie la structure de toutes les solutions positives d une equation de schrodinger stationnaire ou le potentiel v est suppose etre une fonction a valeur reelle appartenant a l p loc r n avec p n 2 pour n 2 et p 1 pour n 1 on considere une formule de representation explicite pour des solutions positives leurs asymptotes a l infini et le probleme de dirichlet avec une condition aux limites a l infini

two particle spectrum of the generator for stochastic model of planar rotators at high temperatures we study two particle spectrum branches of the generator in the stochastic model of planar rotators using the construction of a special basis in two particle invariant subspaces we prove that the branches of the spectrum are in a small neighborhood of the point 2 we prove the existence of two bound states in addition to the continuous part of the spectrum in the one dimensional case

multi parameter local hardy spaces abstract though multi parameter hardy space theory has been well developed in the past half century not much has been studied for a local hardy space theory in the multi parameter settings such multi parameter local hardy spaces can play an important role in studying the boundedness of multi parameter pseudo differential operators multi parameter singular integrals of non convolution type and applications to partial differential equations etc by establishing a bi parameter local reproducing formula bi parameter local hardy space h p r n 1 r n 2 is introduced in this paper this space coincides with l p r n 1 n 2 when p 1 while p 1 this space is substantially different from the classical bi parameter hardy spaces h p r n 1 r n 2 we will establish its atomic decomposition of the bi parameter local hardy spaces and as an application we prove the boundedness from h p r n 1 r n 2 to l p r n 1 n 2 of the bi parameter singular integral operators in inhomogeneous journe class for p near 1 for the simplicity we have chosen to present all the results in the bi parameter setting nevertheless all of them hold for arbitrary number of parameters the multi parameter local theory developed in this paper can serve as a model case for similar theory in other multi parameter settings

calderon zygmund theory for product domains hp spaces this article deals with the boundedness properties of singular integrals that generalize the double hilbert transform on product domains the action of these operators on hardy spaces and on l log l is discussed