uniqueness of gradient gibbs measures with disorder we consider in a uniformly strictly convex potential regime two versions of random gradient models with disorder in model a the interface feels a bulk term of random fields while in model b the disorder enters through the potential acting on the gradients we assume a general distribution on the disorder with uniformly bounded finite second moments it is well known that for gradient models without disorder there are no gibbs measures in infinite volume in dimension d 2 while there are shift invariant gradient gibbs measures describing an infinite volume distribution for the gradients of the field as was shown by funaki and spohn commun math phys 185 1 36 1997 van enter and kulske proved in ann appl probab 18 1 109 119 2008 that adding a disorder term as in model a prohibits the existence of such gradient gibbs measures for general interaction potentials in d 2 in cotar and kulske ann appl probab 22 5 1650 1692 2012 we proved the existence of shift covariant random gradient gibbs measures for model a when d ge 3 the disorder is i i d and has mean zero and for model b when d ge 1 and the disorder has a stationary distribution in the present paper we prove existence and uniqueness of shift covariant random gradient gibbs measures with a given expected tilt u in mathbb r d and with the corresponding annealed measure being ergodic for model a when d ge 3 and the disordered random fields are i i d and symmetrically distributed and for model b when d ge 1 and for any stationary disorder dependence structure we also compute for both models for any gradient gibbs measure constructed as in cotar and kulske ann appl probab 22 5 1650 1692 2012 when the disorder is i i d and its distribution satisfies a poincare inequality assumption the optimal decay of covariances with respect to the averaged over the disorder gradient gibbs measure

delocalization of two dimensional random surfaces with hard core constraints we study the fluctuations of random surfaces on a two dimensional discrete torus the random surfaces we consider are defined via a nearest neighbor pair potential which we require to be twice continuously differentiable on a possibly infinite interval and infinity outside of this interval no convexity assumption is made and we include the case of the so called hammock potential when the random surface is uniformly chosen from the set of all surfaces satisfying a lipschitz constraint our main result is that these surfaces delocalize having fluctuations whose variance is at least of order log n where n is the side length of the torus we also show that the expected maximum of such surfaces is of order at least log n the main tool in our analysis is an adaptation to the lattice setting of an algorithm of richthammer who developed a variant of a mermin wagner type argument applicable to hard core constraints we rely also on the reflection positivity of the random surface model the result answers a question mentioned by brascamp et al on the hammock potential and a question of velenik

nonlinear aspects of the renormalization group flows of dyson s hierarchical model we review recent results concerning the renormalization group rg transformation of dyson s hierarchical model hm this model can be seen as an approximation of a scalar field theory on a lattice we introduce the hm and show that its large group of symmetry simplifies drastically the blockspinning procedure several equivalent forms of the recursion formula are presented with unified notations rigourous and numerical results concerning the recursion formula are summarized it is pointed out that the recursion formula of the hm is inequivalent to both wilson s approximate recursion formula and polchinski s equation in the local potential approximation despite the very small difference with the exponents of the latter we draw a comparison between the rg of the hm and functional rg equations in the local potential approximation the construction of the linear and nonlinear scaling variables is discussed in an operational way we describe the calculation of non universal critical amplitudes in terms of the scaling variables of two fixed points this question appears as a problem of interpolation between these fixed points universal amplitude ratios are calculated we discuss the large n limit and the complex singularities of the critical potential calculable in this limit the interpolation between the hm and more conventional lattice models is presented as a symmetry breaking problem we briefly introduce models with an approximate supersymmetry one important goal of this review is to present a configuration space counterpart suitable for lattice formulations of functional rg equations formulated in momentum space often called exact rg equations and abbreviated erge

frequency operator for anharmonic oscillators by perturbation theory we obtain approximate analytical expressions for the frequency operator for anharmonic oscillators by means of perturbation theory and a unitary transformation

geometrical crossover and self similarity of dla and viscous fingering clusters abstract diffusion limited aggregation dla clusters and the structures observed in viscous fingering experiments at high capillary numbers are tree like fractals the different branches may be assigned a branch order in a way that exhibits scaling and permits a self similar characterization by the bifurcation ratio rn and the length ratio rl of branches of different orders the fractal dimension is given by d log r n log 1 r l good agreement between experiments and simulations is found a crossover function characterizes the branch orders and we conclude that dla is in a state of geometrical crossover branches are linear up to their length l but fractally distributed on length scales much larger than l the effective fractal dimension of dla depends on how d is measured and over what part of the cluster

precise determination of the energy levels of the anharmonic oscillator from the quantization of the angle variable using an ansatz motivated by the classical form of el phi where phi is the angle variable we construct operators which satisfy the commutation relations of the creation annihilation operators for the anharmonic oscillator the matrix elements of these operators can be expressed in terms of entire functions in the position complex plane these functions provide solutions of the ricatti equation associated with the time independent schrodinger equation we relate the normalizability of the eigenstates to the global properties of the flows of this equation these exact results yield approximations which complement the wkb approximation and allow an arbitrarily precise determination of the energy levels we give numerical results for the first 10 levels with 30 digits we address the question of the quantum integrability of the system

exact steady states of disordered hopping particle models with parallel and ordered sequential dynamics a one dimensional driven lattice gas with disorder in the particle hopping probabilities is considered it has previously been shown that in the version of the model with random sequential updating a phase transition occurs from a low density inhomogeneous phase to a high density congested phase here the steady states for both parallel fully synchronous updating and ordered sequential updating are solved exactly the phase transition is shown to persist in both cases with the critical densities being higher than that for random sequential dynamics the steady state velocities are related to the fugacity of a bose system suggesting a principle of minimization of velocity a generalization of the dynamics to the case where the hopping probabilities depend on the number of empty sites in front of the particles is also solved exactly in the case of parallel updating the models have natural interpretations as simplistic descriptions of traffic flow the relation to more sophisticated traffic flow models is discussed

transient dynamics of on line learning in two layered neural networks the dynamics of on line learning in neural networks with continuous units is dominated by plateaux in the time dependence of the generalization error using tools from statistical mechanics we show for a soft committee machine the existence of several fixed points of the dynamics of learning that give rise to complicated behaviour such as cascade like runs through different plateaux with a decreasing value of the corresponding generalization error we find learning rate dependent phenomena such as splitting and disappearing of fixed points of the equations of motion the dependence of plateau lengths on the initial conditions is described analytically and simulations confirm the results

on the absence of breakdown of symmetry for the plane rotator model with long range random interaction we study the plane rotator model with hamiltonian frac 1 2 sum limits x ne y frac jxy cos mathcal o x mathcal o y left x y right 3 mathcal e where jxy for different pair x y are independent symmetric unbounded random variables it is proved that for almost all j all gibbs states p j are invariant by rotation

clustering in low dimensional so n invariant statistical models with long range interactions we obtain upper bounds for the two point correlation functions in statistical models in one or two dimensions which haveso n symmetry this clarifies upper bounds for long range interactions for which there exists clustering

mass generations in two dimensional hierarchical heisenberg model of migdal kadanoff type finiteness of correlation length in the 2d heisenberg model is established within the migdal kadanoff approximate renormalization recursion formulas

damage spreading in a two dimensional potts ferromagnet in an external field we study using the damage spreading method and within the heat bath dynamics the dynamical behaviour of the square lattice three state potts ferromagnet subjected to an external field in the zero field case we verify that this very simple model presents besides the transition consistent with the equilibrium one a new dynamical chaotic phase with unusual features although the unexpected transition occurs within the error bars at the static ising critical temperature it is in the directed percolation universality class we observe that the application of a uniform magnetic field does not destroy any of the two transitions while the unexpected one is annihilated by a new kind of field which plays the role of a conjugate field to the hamming distance

energy entropy inequalities for classical lattice systems a new characterization of equilibrium states for classical lattice systems is given in terms of correlation inequalities their physical meaning is found to express thermodynamic stability we demonstrate the applicability of the inequalities in specific models

orientation of particle attachment and local isotropy in diffusion limited aggregation dla we simulate 50 off lattice diffusion limited aggregation dla clusters of one million particles each the probability distribution of the angle of attachment of arriving particles with respect to the local radial direction is obtained numerically for increasing cluster size n the distribution crosses over extremely accurately to a cosine with an amplitude which decreases towards zero as a power law in n from this specific viewpoint asymptotically large dla clusters are locally isotropic

completeness of coherent states associated with self similar potentials and ramanujan s integral extension of the beta function a decomposition of identity is given as a complex integral over the coherent states associated with a class of shape invariant self similar potentials there is a remarkable connection between these coherent states and ramanujan s integral extension of the beta function

comment on a theorem of hojman and its generalizations an invariant derived using hojman s conservation law is shown to be trivial for all noether symmetries

strict convexity of the free energy for a class of non convex gradient models we consider a gradient interface model on the lattice with interaction potential which is a non convex perturbation of a convex potential we show using a one step multiple scale analysis the strict convexity of the surface tension at high temperature this is an extension of funaki and spohn s result 8 where the strict convexity of potential was crucial in their proof

damping optimization in mechanical systems with external force abstract we consider a mechanical system excited by external force model of such a system is described by the system of ordinary differential equations m x t d x t kx t f t where matrices m k mass and stiffness are positive definite and the vector f corresponds to an external force the damping matrix d is assumed to be positive semidefinite and has a small rank we introduce two criteria that allow damping optimization of mechanical system excited by an external force since in general a damping optimization is a very demanding problem we provide a new formulas which have been used for efficient damping optimization the efficiency of new formulas is illustrated with a numerical experiment

on the relative entropy fora any subset of the bounded operators on a hilbert space containing the unit and and restrictions of states on toa ent a the entropy of relative to given the information ina is defined and given an axiomatic characterisation it is compared with ent the relative entropy introduced by umegaki and generalised by various authors which is defined only fora an algebra it is proved that ent and ent s agree on pairs of normal states on an injective von neumann algebra it is also proved that ent always has all the most important properties known for ent s monotonicity concavity w upper semicontinuity etc

on equivalence of spin and field pictures of lattice systems we investigate the spin and field systems on a lattice connected by the kac siegert transform it is shown that the structures of corresponding theories are equivalent in the sense of isomorphy of space of gibbs states and order parameters using the idea of equivalence of spin and field pictures we exhibit a class of lattice systems possessing infinitely uncountably many ground states the systems of this type with infinite range slow decaying interactions are expected to have a spin glass phase transition

spectral gap and decay of correlations in u 1 symmetric lattice systems in dimensions d 2 we consider many body systems with a global u 1 symmetry on a class of lattices with the fractal dimension d 2 and their zero temperature correlations whose observables behave as a vector under the u 1 rotation for a wide class of the models we prove that if there exists a spectral gap above the ground state then the correlation functions have a stretched exponentially decaying upper bound this is an extension of the mcbryan spencer method at finite temperatures to zero temperature the class includes quantum spin and electron models on the lattices and our method also allows finite or infinite quasi degeneracy of the ground state the resulting bounds rule out the possibility of the corresponding magnetic and electric long range orders

large deviation principle for one dimensional vector spin models with kac potentials we consider the one dimensional planar rotator and classical heisenberg models with a ferromagnetic kac potential j r j yr j with compact support below the lebowitz penrose critical temperature the limit mean field theory exhibits a phase transition with a continuum of equilibrium states indexed by the magnetization vectors m s s any unit vector and m the curie weiss spontaneous magnetization we prove a large deviation principle for the associated gibbs measures then we study the system in the limit 0 below the above critical temperature we prove that the norm of the empirical spin average in blocks of order 1 converges to m uniformly in intervals of order p for any p 1 we also give a lower bound to the scale on which the change of phase occurs by showing that the empirical spin average is approximately constant on intervals having length of order 1 with 0 1 small enough

the general structure of eigenvalues in nonlinear oscillators hilbert spaces of bounded one dimensional nonlinear oscillators are studied it is shown that the eigenvalue structure of all such oscillators have the same general form they depend only on the ground state energy of the system and a single function h of the hamiltonian operator h it is also found that the hilbert space of the nonlinear oscillator is unitarily inequivalent to the hilbert space of the simple harmonic oscillator providing an explicit example of haag s theorem a number operator for the nonlinear oscillator is constructed and the general form of the partition function and average energy of a nonlinear oscillator in contact with a heat bath is determined connection with the wkb result in the semiclassical limit is made the analysis is then applied to the case of the x4 anharmonic oscillator as an explicit example

damping optimization over the arbitrary time of the excited mechanical system in this paper we consider damping optimization in mechanical system excited by an external force we use optimization criteria based on minimizing average energy amplitude and average displacement amplitude over the arbitrary time as the main result we derive explicit formulas for objective functions these formulas can be implemented efficiently and accelerate optimization process significantly which is illustrated in a numerical example

quantum dynamics of parasupersymmetric and shape invariant coupled systems a class of bound state problems which represents the coupling of a three level atom with a two dimensional system involving two shape invariant potentials was introduced in a previous paper in this paper we considered second order parasupersymmetric quantum mechanical models and two possible kinds for the coupling hamiltonian linear and nonlinear in the potential ladder operators in the present paper using an algebraic formulation for shape invariant potential systems we study the quantum dynamics of these coupled systems and obtain the temporal behaviour of some dynamical variables related with the atom and the coupling potentials an application is given for a couple of shape invariant potentials poschl teller self similar potentials

properties of the scalar universal equations the variational properties of the scalar so called universal equations are reviewed and generalized in particular we note that each member of the euler hierarchy may have an explicit field dependence the euler hierarchy itself is given a new interpretation in terms of the formal complex of variational calculus and is shown to be related to the algebra of variational symmetries of the first source form

the global markov property for equilibrium states which are determined by correlations in a strip consider an interaction round face potential on the lattice d which may include nearest neighbor and next nearest neighbor pair interactions as well as more general plaquette terms assuming some periodicity of the potential it is shown that under the condition that equilibrium states can be distinguished by expectation values of sufficiently local observables equilibrium states possess a global markov property the condition under which this markov property is shown to hold is met in particular in case the equilibrium state is unique or determined by the magnetization the proof is based on an application of the variational principle to states which are constructed like markov chains

on the absence of spontaneous symmetry breaking and of crystalline ordering in two dimensional systems we develop a unified approach based on araki s relative entropy concept to proving absence of spontaneous breaking of continuous internal symmetries and translation invariance in two dimensional statistical mechanical systems more precisely we show that under rather general assumptions on the interactions all equilibrium states of a two dimensional system have all the symmetries compact internal and spatial of the dynamics except possibly rotation invariance rotation invariance remains unbroken if connected correlations decay more rapidly than the inverse square distance we also prove that two dimensional systems with a non compact internal symmetry group like anharmonic crystals typically do not have gibbs states

anharmonic oscillator constructing the strong coupling expansions two novel approaches to construction of the strong coupling expansion for the anharmonic oscillator with the potential v x 1 2 x2 g 4 x4 are proposed the first one is simply a straightforward solution of the schrodinger equation via the nonlinearization technique resulting in the rapidly convergent perturbation series the second one is a version of the path integral perturbation theory but with an unconventional choice of the zeroth approximation action nine leading coefficients of the strong coupling expansion are computed they decrease rapidly the ninth one being of the order of 10 9 three leading corrections of the nonlinearization approach provide the ground state energy within a relative accuracy of 10 7 10 9 at an arbitrary coupling g the explicit formulas for corrections enable one to study the analyticity properties of the energy as a function of the coupling g in the second approach the strong coupling expansion coefficients are computed as an infinite linear series of the weak

a new proof of ruelle s superstability bounds for a slightly stronger assumptions on the interaction we give a very transparent proof of ruelle s result in the language of poisson integral measure representation for the correlation functions on the configuration space using some kind of cluster expansion in the densities of configurations

u 1 su 2 gauge invariance of non relativistic quantum mechanics and generalized hall effects we show that the non relativistic quantum mechanics of particles with spin coupled to an electromagnetic field has a naturalu 1 su 2 gauge invariance ward identities reflecting this gauge invariance combined with an assumption of incompressibility of a system of such particles in an appropriate external field and for suitable values of the particle density permit us to determine the form of the effective action of the system as a functional of small fluctuations in the electromagnetic field in the large distance adiabatic limit in this limit the action is found to have a universal form we present explicit results for two dimensional incompressible electron fluids and apply them to derive the equations of linear response theory describing a variety of generalized hall effects sum rules for the hall conductivities magnetic susceptibilities and other quantities of physical interest are found

lie symmetry reductions and exact solutions of some differential difference equations in this paper a class of differential difference equations ddes are considered for lie group analysis with the help of symbolic computation mathematica the continuous lie point symmetry technique is extended to obtain corresponding infinitesimals similarity reductions are derived by solving the characteristic equations then some exact solutions are presented by using inverse transformations in addition starting from concrete realization of the generalized virasoro type symmetry algebra f1 f2 f 1f2 f1f 2 many high dimensional ddes can be derived for example we give out the 2 1 dimensional toda lattice modified toda lattice and special toda lattice in a uniform way

on the fourier gauss transforms of some q exponential and q trigonometric functions we examine the properties of q exponential and q trigonometric functions recently introduced and discussed in the literature it is shown that they are related to jackson s q analogues of the exponential and trigonometric functions by classical fourier gauss transformations

correlation and response in the backgammon model the ehrenfest legacy we pursue our investigation of the non equilibrium dynamics of the backgammon model a dynamical urn model which exhibits aging and glassy behaviour at low temperature we present an analytical study of the scaling behaviour of the local correlation and response functions of the density fluctuations of the model and of the associated fluctuation dissipation ratios throughout the regime of low temperature and long times this analysis includes the aging regime the convergence to equilibrium and the crossover behaviour between them

symmetry techniques for the al salam chihara polynomials an algebraic interpretation for the al salam chihara q orthogonal polynomials is presented using an explicit realization in terms of divided difference operators their symmetry q algebra is determined and studied this algebraic structure allows one to derive a new expansion formula for these polynomials

on the symmetry of the gibbs states in two dimensional lattice systems under fairly general conditions if a two dimensional classical lattice system has an internal symmetry groupg which is a compact connected lie group then all gibbs states areg invariant

euclidean invariance in statistical mechanics of classical continuous system it is shown that equilibrium states of classical particles with short range interactions are euclidean invariant whenever their correlation functions have a clustering which is integrable the relation between invariance and clustering is analysed for spatial rotations and internal rotational degrees of freedom the analysis is then extended to the case of long range interactions including the coulomb force and jellium systems

large deviations and variational principle for harmonic crystals we consider massless gaussian fields with covariance related to the green function of a long range random walk on d these are viewed as gibbs measures for a linear quadratic interaction we establish thermodynamic identities and prove a version of gibbs variational principle showing that translation invariant gibbs measures are characterized as minimizers of the relative entropy density we then study the large deviations of the empirical field of a gibbs measure we show that a weak large deviation principle holds at the volume order with rate given by the relative entropy density

the mermin wagner phenomenon and cluster properties of one and two dimensional systems we give optimal conditions concerning the range of interactions for the absence of spontaneous breakdown of continuous symmetries for one and two dimensional quantum and classical lattice and continuum systems for a class of models verifying infrared bounds our conditions are necessary and sufficient using the same techniques we obtain a priori bounds on clustering for systems with continuous symmetry improving results of jasnow and fisher

some properties of correlations of quantum lattice systems in thermal equilibrium simple proofs of uniqueness of the thermodynamic limit of kms states and of the decay of equilibrium correlations are presented for a large class of quantum lattice systems at high temperatures new quantum correlation inequalities for general heisenberg models are described finally a simplified derivation of a general result on power law decay of correlations in 2d quantum lattice systems with continuous symmetries is given extending results of mcbryan and spencer for the 2d classical xy model

the construction of some important classes of generalized coherent states the nonlinear coherent states method considering some important classes of generalized coherent states known in the literature we demonstrated that all of them can be created via conventional methods i e the lowering operator eigenstate and the displacement operator techniques using the nonlinear coherent states approach as a result we obtained a unified method to construct a large class of coherent states which have already been introduced by different prescriptions

local isotropy and geometry of particle flux lines in diffusion limited aggregation we present a theoretical description of the branch orientation and the geometry of walker flux lines in diffusion limited aggregation dla it is based on the self similarity and non directedness of the branch structure and explains recent numerical results on the probability distribution of the branch orientation and the angle of particle attachment our results support the idea that asymptotically large off lattice dla clusters are locally isotropic and every individual walker flux line exhibits a certain type of self similarity

relative hamiltonians which are not bounded from above abstract let be a normal state on a von neumann algebra a and let h be a self adjoint operator affiliated with a which is bounded below we discuss the perturbed state h h is defined when it exists to be the unique state maximizing the function ent a h where ent a is the entropy of relative to we prove that given a state with x for some x r then there exists a generalized operator h with h we prove that h h h h whenever the right hand side exists we prove that if is faithful and h h for which h is a kms state with respect to the modular group of is a dense subset of the set of states kms with respect to the modular group of h

lyapunov optimization of a damped system our aim is to optimize the damping of a linear vibrating system the penalty function is the average total energy which is equal to the trace of the corresponding lyapunov solution we prove the existence and the uniqueness of the global minimum if the damping varies over the set of all possible positive definite matrices the minimum is shown to be taken on the so called modal critical damping thus confirming a long existing conjecture we also give some preliminary results concerning dampings which depend linearly on the viscosity parameters whereas the damper positions are kept fixed we produce physical examples on which the minimum is taken on a negative viscosity or which have several local minima both phenomena seem to be a consequence of a bad choice of the damper positions

absence of phase transitions in certain one dimensional long range random systems an ising chain is considered with a potential of the formj i j i j where thej i j are independent random variables with mean zero the chain contains both randomness and frustration and serves to model a spin glass a simple argument is provided to show that the system does not exhibit a phase transition at a positive temperature if 1 this is to be contrasted with a ferromagnetic interaction which requires 2 the basic idea is to prove that the surface free energy between two half lines is finite although the surface energy may be unbounded ford dimensional systems it is shown that the free energy does not depend on the specific boundary conditions if 1 2 d

perturbation theory free from secular terms for the equations of motion of anharmonic oscillators we develop a time independent perturbation theory in operator form for the equations of motion of classical and quantum mechanical anharmonic oscillators the method focuses on the frequency of the motion in the former case and in a frequency operator in the latter producing a power series in a small coupling constant the resulting expressions are free from secular instabilities that commonly appear in perturbation treatments of nonlinear problems

lie bialgebra quantizations of the oscillator algebra and their universal r matrices all coboundary lie bialgebras and their corresponding poisson lie structures are constructed for the oscillator algebra generated by quantum oscillator algebras are derived from these bialgebras by using the lyakhovsky and mudrov formalism and for some cases quantizations at both algebra and group levels are obtained including their universal r matrices

calculation of energy eigenvalues for the quantum anharmonic oscillator with a polynomial potential we choose the squeezed vacuum state as a one parameter trial wavefunction to minimize the energy of an anharmonic oscillator with a polynomial perturbation potential using the optimal vacuum state obtained we generate the optimal generalized number state basis and set up the hamiltonian in the normal ordered form then truncating the basis we produce finite dimensional matrices in a simple manner and find their eigenvalues by standard methods we will apply our method to the morse potential whose exact eigenvalues are known as a check on the accuracy of the results that we have derived two models of the double well and the quartic potential are considered as examples

quantum energy entropy inequalities a new method for proving the absence of symmetry breaking for quantum systems we develop a new method based on a general energy entropy inequality to rule out spontaneous breaking of symmetries the main advantage of our scheme consists in its clear cut physical significance and its new areas of applicability in particular we can handle discrete symmetry groups as well as continuous ones finally a few illustrations are discussed

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

arbitrarily accurate eigenvalues for one dimensional polynomial potentials we show that the riccati form of the one dimensional schrodinger equation can be reformulated in terms of two linear equations depending on an arbitrary function g when g and the potential as for anharmonic oscillators are polynomials the solutions of these two equations are entire functions l and k and the zeros of k are identical to those of the wavefunction requiring such a zero at a large but finite value of the argument yields low energy eigenstates with exponentially small errors approximate formulae for these errors are provided we explain how to choose g in order to dramatically improve the numerical treatment the method yields many significant digits with modest computer means we discuss the extension of this method in the case of several variables

a mermin wagner theorem for gibbs states on lorentzian triangulations we establish a mermin wagner type theorem for gibbs states on infinite random lorentzian triangulations lt arising in models of quantum gravity such a triangulation is naturally related to the distribution p of a critical galton watson tree conditional upon non extinction at the vertices of the triangles we place classical spins taking values in a torus m of dimension d with a given group action of a torus g of dimension d d in the main body of the paper we assume that the spins interact via a two body nearest neighbor potential u x y invariant under the action of g we analyze quenched gibbs measures generated by u and prove that for p almost all lorentzian triangulations every such gibbs measure is g invariant which means the absence of spontaneous continuous symmetry breaking

on estimating the derivatives of symmetric diffusions in stationary random environment with applications to ϕ interface model we consider diffusions on d or random walks on d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients we show existence and holder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives in the case of random walks these estimates are applied to the ginzburg landau interface model

recurrent random walks and the absence of continuous symmetry breaking on graphs we consider geometrically disordered systems with a continuous symmetry groupg where the internal degrees of freedom are attached to the vertices of a graph we show that equilibrium states remaing invariant at any temperaturet 0 if a random walk on the graph is recurrent this generalizes a previous result obtained by cassi

generalized coherent squeezed and intelligent states for exactly solvable quantum systems and the analogue of the displacement and squeezing operators using an algebraic approach we construct purely coherent and squeezed states for a set of exactly solvable quantum confined systems assuming a generalization of both the ladder operator and the displacement operator methods showing the equivalence between the two procedures we obtain the analogue of the displacement and the squeeze operators and study their basic relations and their connections with the properties of the purely coherent and squeezed states that they produce in a second approach we construct generalized squeezed coherent and intelligent states for these systems considering the successive action of the displacement and the squeeze operators in the first case and the combined action of these operators in the second case we apply the generalized approach for shape invariant systems with potential parameters related by a translation and by a scaling and show that the presence of the generalizing functionals of the potential parameters in the formalism opens a large number of possibilities in the construction of these special quantum states

frequency operator for anharmonic oscillators we develop a simple approach based on time independent perturbation theory that leads to a frequency operator for quantum mechanical models this method is suitable for the calculation of time dependent physical properties free from secular terms in particular we obtain the first order perturbation correction to the frequency operator for a class of anharmonic oscillators

phase structure of two dimensional spin models and percolation for a class of classical spin models in 2d satisfying a certain continuity constraint it is proven that some of their correlations do not decay exponentially the class contains discrete and continuous spin systems with abelian and non abelian symmetry groups for the discrete models our results imply that they show either long range order or are in a soft phase characterized by powerlike decay of correlations for the continuous models only the second possibility exists the continuous models include a version of the plane rotator o 2 model for this model we rederive modulo two conjectures the frohlich spencer result on the existence of the kosterlitz thouless phase in a very simple way the proof is based on percolation theoretic and topological arguments

another addition theorem for the q exponential function an addition formula with respect to a variable and parameter is established for the basic exponential function on a q quadratic grid

dromion like structures in a 3 1 dimensional kdv type equation dromion structures have been found in many 2 1 dimensional models the similar structure in 3 1 dimensions is studied in this paper for a kdv type equation starting from the bilinear form of the model we found that there are five types of multi dromion solutions for its potentials say the first type of multi dromion solution is driven by multi camber solitons with one of them being non parallel to the x axis the second third and fourth types of multi dromion solutions are driven by two camber solitons and one two and three sets of parallel plane solitons respectively the fifth type of multi dromion solution is driven by one set of parallel plane solitons one set of camber solitons which are parallel to the x axis and one camber soliton which is non parallel to the x axis phase shifts may be involved in the interactions among the multi dromions for the first and fifth types of solutions a single dromion solution of the model may possess quite a free shape for instance the point like dromions ring type dromions extended and sharp dromions and oscillatory dromions can be obtained by selecting some arbitrary functions appropriately

asymptotic iteration method for eigenvalue problems an asymptotic iteration method for solving second order homogeneous linear differential equations of the form y 0 x y s0 x y is introduced where 0 x 0 and s0 x are c functions applications to schrodinger type problems including some with highly singular potentials are presented