coherent state maps related to the bounded positive operators we show that for any bounded positive operator h with the simple spectrum one can canonically define two coherent state maps the algebras generated by annihilation operators defined by these coherent state maps are studied we describe also how the toda isospectral deformation of h deforms the corresponding coherent state maps and the related operator algebras

uniform boundary harnack principle and generalized triangle property let d be a bounded open subset in rd d 2 and let g denote the green function for d with respect to 2 0 2 d if 2 assume that d satisfies the interior corkscrew condition if 2 i e if g is the classical green function on d assume more restrictively that d is a uniform domain let g g y0 1 for some y0 d based on the uniform boundary harnack principle it is shown that g has the generalized triangle property which states that g z y g z cg x y g x when d z x d z y an intermediate step is the approximation g x y x y dg x g y g a 2 where a is an arbitrary point in a certain set b x y this is discussed in a general setting where d is a dense open subset of a compact metric space satisfying the interior corkscrew condition and g is a quasi symmetric positive numerical function on d d which has locally polynomial decay and satisfies harnack s inequality under these assumptions the uniform boundary harnack principle the approximation for g and the generalized triangle property turn out to be equivalent

intrinsic ultracontractivity and small perturbation for one dimensional generalized diffusion operators abstract we consider a one dimensional generalized diffusion operator g on an open interval which is not necessarily bounded we characterize sp i e 1 is a small perturbation of the operator g in terms of the classification of the boundary points from the view point of diffusion processes and give a sufficient condition for iu i e the associated heat kernel is intrinsically ultracontractive

new q deformed coherent states with an explicitly known resolution of unity we construct a new family of q deformed coherent states zq where 0 q 1 these states are normalizable on the whole complex plane and continuous in their label z they allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated stieltjes power moment problem and expressed in terms of one of the two jacksons s q exponentials they also permit exact evaluation of matrix elements of physically relevant operators we use this to show that the photon number statistics for the states is sub poissonian and that they exhibit quadrature squeezing as well as an enhanced signal to quantum noise ratio over the conventional coherent state value finally we establish that they are the eigenstates of some deformed boson annihilation operator and study some of their characteristics in deformed quantum optics

critical phenomena in linear elliptic problems abstract in this article we consider the existence and nonexistence of principal eigenvalues for the linear elliptic equation u v x u in r n and u d 1 2 r n where n 3 and v l 1 loc r n with v 0 in r n we establish that when potential v has strong singularities phenomena which appear under critical exponent nonlinearities brezis and nirenberg are present in linear problems as well

intrinsic ultracontractivity of a schrodinger semigroup in rn abstract we give a possibly sharp sufficient condition on the electric potential q r n 0 in the schrodinger operator a q x on l 2 r n that guarantees that the schrodinger heat semigroup e a t t 0 on l 2 r n generated by a is intrinsically ultracontractive moreover if q x q x is radially symmetric we show that our condition on q is also necessary i e truly sharp it reads r 0 q r 1 2 d r for some r 0 0 our proofs make essential use of techniques based on a logarithmic sobolev inequality rosen s inequality proved via a new fenchel young inequality and a very precise asymptotic formula due to hartman and wintner

integrable multi boson systems and orthogonal polynomials the strict relation between a certain class of multi boson hamiltonian systems and the corresponding class of orthogonal polynomials is established the correspondence is effectively used to integrate the systems as an explicit example we integrate the class of multi boson systems corresponding to q hahn class polynomials

a ground state alternative for singular schrodinger operators let a be a quadratic form associated with a schrodinger operator l a v on a domain r d i fa is nonnegative on c 0 then either there is w 0 such that

representations of the quadratic algebra and partially asymmetric diffusion with open boundaries we consider the one dimensional partially asymmetric exclusion model with open boundaries the model describes a system of hard core particles that hop stochastically in both directions with different rates at both boundaries particles are injected and extracted by means of the method of derrida et al the stationary probability measure can be expressed as a matrix product state involving two matrices forming a fock like representation of a general quadratic algebra we obtain the representations of this algebra which were unknown in the mathematical literature and use the two dimensional one to derive exact expressions for the density profile and correlation functions using the correspondence between the stochastic model and a quantum spin chain we obtain exact correlation functions for a spin heisenberg xxz chain with non diagonal boundary terms generalizations to other reaction diffusion models are discussed

large sets at infinity and maximum principle on unbounded domains for a class of sub elliptic operators abstract maximum principles on unbounded domains play a crucial role in several problems related to linear second order pdes of elliptic and parabolic type in this paper we consider a class of sub elliptic operators l in r n and we establish some criteria for an unbounded open set to be a maximum principle set for l we extend some classical results related to the laplacian by deny hayman and kennedy and to the sub laplacians on stratified lie groups by bonfiglioli and the second named author

nonlinear equations and weighted norm inequalities we study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities in particular we obtain explicit criteria for the solvability of the dirichlet problem u v u w u 0 on u 0 on on a regular domain in rn in the superlinear case q 1 the coefficients v w are arbitrary positive measurable functions or measures on we also consider more general nonlinear differential and integral equations and study the spaces of coefficients and solutions naturally associated with these problems as well as the corresponding capacities our characterizations of the existence of positive solutions take into account the interplay between v w and the corresponding green s kernel they are not only sufficient but also necessary and are established without any a priori regularity assumptions on v and w we also obtain sharp two sided estimates of solutions up to the boundary some of our results are new even if v 1 and is a ball or half space the corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so called 3g inequality by an elementary integration by parts argument this also gives a new unified proof for some classical inequalities including the carleson measure theorem for poisson integrals and trace inequalities for riesz potentials and green potentials

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

a realization of generalized deformed oscillators with cyclic symmetry the algebra of generalized deformed oscillators that possess cyclic symmetry is studied the general set of multilinear operator relations is presented which characterizes the multilinear part of the algebra of cyclic extended deformed oscillators the realization scheme of generalized deformed oscillators with cyclic symmetry in terms of those without cyclic symmetry is established it is shown that all generalized deformed oscillators with cyclic symmetry constructed by the scheme are actually equivalent in the viewpoint of mathematics though each generalized deformed oscillator admits its own particular cyclic extended counterpart examples are given to illustrate this peculiar realization scheme

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

on the picard principle for δ μ given a local kato measure on mathbb r d setminus 0 d ge 2 let mathcal h 0 delta mu u be the convex cone of all continuous real solutions u 0 to the equation u u 0 on the punctured unit ball u satisfying lim x to 1 u x 0 it is shown that mathcal h 0 delta mu u ne 0 if and only if the operator f mapsto int u g cdot y f y d mu y where g denotes the green function on u is bounded on mathcal l 2 u mu and has a norm which is at most one moreover extremal rays in mathcal h 0 delta mu u are characterized and it is proven that satisfies the picard principle on u that is that mathcal h 0 delta mu u consists of one ray provided there exists a suitable sequence of shells in u such that on these shells is either small or not too far from being radial further it is shown that the verification of the picard principle can be localized several results on l2 sub eigenfunctions and 3g inequalities which are used in the paper but may be of independent interest are proved at the end of the paper

on criticality and ground states of second order elliptic equations ii abstract in this article we examine a necessary and sufficient condition for the existence of a positive solution of the equation p tw u 0 in r n where p is a second order alliptic operator which is critical in w c 0 and t 0 is small enough

on the asymptotic decay of l2 solutions of one body schrodinger equations in unbounded domains the asymptotic decay of l 2 solutions of schrodinger equations v 0 in r x er n x r r is investigated where v x v 1 r v 2 x with v 1 for r and with some e 0 for large r under additional assumptions on the decay of v 1 pointwise upper bounds to and lower bounds to the spherical average of are given showing the same asymptotics for r for the case v const 0 for r investigated in 8 a simplified treatment is given

semismall perturbations semi intrinsic ultracontractivity and integral representations of nonnegative solutions for parabolic equations we consider nonnegative solutions of a parabolic equation in a cylinder d i where d is a noncompact domain of a riemannian manifold and i 0 t with 0 t or i 0 under the assumption ssp i e the constant function 1 is a semismall perturbation of the associated elliptic operator on d we establish an integral representation theorem of nonnegative solutions in the case i 0 t any nonnegative solution is represented uniquely by an integral on d 0 md 0 t where md is the martin boundary of d for the elliptic operator and in the case i 0 any nonnegative solution is represented uniquely by the sum of an integral on md 0 and a constant multiple of a particular solution we also show that ssp implies the condition siu i e the associated heat kernel is semi intrinsically ultracontractive

time quantization and q deformations we extend to quantum mechanics the technique of stochastic subordination by means of which one can express any semi martingale as a time changed brownian motion as examples we considered two versions of the q deformed harmonic oscillator in both ordinary and imaginary time and show how these various cases can be understood as different patterns of time quantization rules

martin boundaries of elliptic skew products semismall perturbations and fundamental solutions of parabolic equations abstract we consider positive solutions of elliptic partial differential equations on non compact domains of riemannian manifolds we explicitly determine martin boundaries and martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short long time estimates for fundamental solutions of parabolic equations

pointwise eigenfunction estimates and intrinsic ultracontractivity type properties of feynman kac semigroups for a class of levy processes we introduce a class of l evy processes subject to specific regularity conditions and consider their feynman kac semigroups given under a kato class potential using new techniques first we analyze the rate of decay of eigenfunctions at infinity we prove bounds on subaveraging functions from which we derive two sided sharp pointwise estimates on the ground state and obtain upper bounds on all other eigenfunctions next by using these results we analyze intrinsic ultracontractivity and related properties of the semigroup refining them by the concept of ground state domination and asymptotic versions we establish the relationships of these properties derive sharp necessary and sufficient conditions for their validity in terms of the behavior of the l evy density and the potential at infinity define the concept of borderline potential for the asymptotic properties and give probabilistic and variational characterizations these results are amply illustrated by key examples

three dimensional quantum groups from contractions of su 2 q contractions of lie algebras and of their representations are generalized to define new quantum groups an explicit and complete exposition is made for the one dimensional heisenberg h 1 q and the two dimensional euclidean quantum group e 2 q obtained by contracting su 2 q

norm estimate of green operator perturbation of green function and integrability of superharmonic functions an operator norm estimate of the green operator for the laplace equation on an open set in the euclidean space is given in terms of capacity this applies to the perturbation of the green function for the schrodinger equation a sufficient condition for the green function for the schrodinger equation to have decay near the boundary comparable to that for the laplace equation is given in terms of an integrable condition for the potential of the schrodinger equation this has an application to the martin boundary of a smooth domain with respect to the schrodinger equation also it is related to the cranston mcconnell inequality about the life time estimate and the integrability of nonnegative superharmonic functions 1991 mathematics subject classification 31a35 31b35 31c35

obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory branched covers of orbit cylinders are the basic examples of holomorphic curves studied in symplectic field theory since all curves with fredholm index one can never be regular for any choice of cylindrical almost complex structure we generalize the obstruction bundle technique of taubes for determining multiple cover contributions from gromov witten theory to the case of moduli spaces with boundary our result proves that the differential in rational symplectic field theory and contact homology is strictly decreasing with respect to the natural action filtration

gravitational descendants in symplectic field theory it was pointed out by y eliashberg in his icm 2006 plenary talk that the rich algebraic formalism of symplectic field theory leads to a natural appearance of quantum and classical integrable systems at least in the case when the contact manifold is the prequantization space of a symplectic manifold in this paper we generalize the definition of gravitational descendants in sft from circle bundles in the morse bott case to general contact manifolds after we have shown using the ideas in okounkov and pandharipande ann math 163 2 517 560 2006 that for the basic examples of holomorphic curves in sft that is branched covers of cylinders over closed reeb orbits the gravitational descendants have a geometric interpretation in terms of branching conditions we follow the ideas in cieliebak and latschev http arixiv org abs 0706 3284v2 math s6 2007 to compute the corresponding sequence of poisson commuting functions when the contact manifold is the unit cotangent bundle of a riemannian manifold

the ptolemy and zbăganu constants of normed spaces abstract in every inner product space h the ptolemy inequality holds the product of the diagonals of a quadrilateral is less than or equal to the sum of the products of the opposite sides in other words x y z w x z y w z y x w for any points w x y z in h it is known that for each normed space x there exists a constant c such that for any w x y z x we have x y z w c x z y w z y x w the smallest such c is called the ptolemy constant of x and is denoted by c p x we study the relationships between this constant and the geometry of the space x and hence with metric fixed point theory in particular we relate the ptolemy constant c p to the zb ganu constant c z and prove that if x is a banach space with c z x 1 3 2 then x has uniform normal structure and therefore the fixed point property for nonexpansive mappings we derive general lower and upper bounds for both c p and c z and calculate the precise values of these two constants for several normed spaces we also present a number of conjectures and open problems

relations for clebsch gordan and racah coefficients in suq 2 and yang baxter equations relations are exploited among clebsch gordan cg and racah coefficients in the algebra suq 2 known as a deformation of su 2 these are used to show that the yang baxter yb relation for the irf interaction round a face model results from one of the symmetry relations for the 9 j symbol specific to suq 2 and that in an asymptotic limit this yb relation becomes the yb relation for the two dimensional vertex model the racah coefficient which has a particularly simple dependence on q is efficiently used such that an asymptotic limit of the racah coefficient is the cg coefficient and another limit gives the factorized s matrix of the vertex model

global comparison of perturbed green functions given a green function g e g with respect to 2 0 2 on a region x where g has a generalized triangle property and given a g bounded signed measure on x necessary and sufficient conditions are given for the existence of a perturbed green function which is comparable with g this is done in the general setting of measurable spaces applications to c1 1 regions in rd and for the classical case 2 to finitely connected regions in 2 are given

a probabilistic approach to bounded positive solutions for schrodinger operators with certain classes of potentials consider the equation 1 2 u vu 0 in r d for d 3 for certain classes of potentials v we use probabilistic tools to study the bounded solutions and the positive solutions for a primary motivation is to offer probabilistic intuition for the results

on positive solutions of second order elliptic equations stability results and classification soit p un operateur uniformement elliptique d ordre 2 defini dans un domaine cr n soit p le cone convexe de toutes les solutions positives de l equation pu 0 dans on etudie les proprietes generales de p pour p et generaux

criticality and ground states for second order elliptic equations abstract let p be a second order elliptic operator in a domain r n suppose that p admits a positive minimal green function g p in it is the main purpose of this paper to prove that if w c is a small perturbation of p in then max w x 0 min w x 0 t u u g p outside a compact set in

maximum and anti maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations in this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations we continue the study of some different notions of small perturbations and discuss their relations to comparisons of green s functions refined maximum and anti maximum principles ground state and the decay of eigenfunctions in particular we show that if v is a positive function which is a semismall perturbation of a subcritical schrodinger operator h defined on a domain omega subset mathbb r d and phi k k geq 0 are the dirichlet eigenfunctions of the equation hu lambda vu then for any k geq 0 the function phi k phi 0 is bounded and has a continuous extension up to the martin boundary of the pair omega h where phi 0 is the ground state of h with a principal eigenvalue lambda 0

a pointwise lower bound for positive solutions of a schrodinger equation in rn abstract strong positivity of the bounded inverse a 1of a schrodinger operator a q x inl2 r n is proved in the following form if a u f 0 inl2 r n withf 0 thenu c 1a e in r n here 1is the positive eigenfunction associated with the principal eigenvalue 1of a andcis a positive constant it is shown that this result is valid if and only if the potentialq x which is assumed to be strictly positive and locally bounded has a sufficiently fast growth as x this result is applied to linear and nonlinear elliptic boundary value problems in strongly ordered banach spaces whose positive cone is generated by the eigenfunction 1 in particular problems of existence and uniqueness are addressed

the q t gaussian process the q t fock space fq t h introduced in this paper is a deformation of the q fock space of bozejko and speicher the corresponding creation and annihilation operators now satisfy the commutation relation aq t f aq t g qaq t g aq t f f g htn turn mathjax off a defining relation of the chakrabarti jagannathan deformed quantum oscillator algebra the moments of the deformed gaussian element sq t h aq t h aq t h are encoded by the joint statistics of crossings and nestings in pair partitions the q 0 t specialization yields a natural single parameter deformation of the full boltzmann fock space of free probability with the corresponding semicircular measure variously encoded via the rogers ramanujan continued fraction the t airy function the t catalan numbers of carlitz riordan and the first order statistics of the reduced wigner process

ground state positivity negativity and compactness for a schrodinger operator in rn abstract we treat the schrodinger operator a q x on l 2 r n with the potential q r n q 0 bounded below and satisfying some reasonable hypotheses on the growth at infinity faster than x 2 as x we are concerned primarily with the compactness of the resolvent a i 1 of a as an operator on the banach space x x f l 2 r n f l r n f x ess sup r n f where denotes the ground state for a if is the ground state energy for a we show that the restricted operator a i 1 x x is not only bounded but also compact for in particular the spectra of a in l 2 r n and x coincide each eigenfunction belongs to x as another consequence we obtain a maximum and an anti maximum principles

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

integral representations of nonnegative solutions for parabolic equations and elliptic martin boundaries abstract we consider nonnegative solutions of a parabolic equation in a cylinder d 0 t where d is a noncompact domain of a riemannian manifold under the assumption iu i e the associated heat kernel is intrinsically ultracontractive we establish an integral representation theorem any nonnegative solution is represented uniquely by an integral on d 0 m d 0 t where m d is the martin boundary of d for the associated elliptic operator we apply it in a unified way to several concrete examples to explicitly represent nonnegative solutions we also show that iu implies the condition sp i e the constant function 1 is a small perturbation of the elliptic operator on d

quantum algebras and q special functions related to coherent states maps of the disc the quantum algebras generated by the coherent states maps of the disc are investigated it is shown that the analytic realization of these algebras leads to a generalized analysis which includes standard analysis as well as q analysis the applications of the analysis to star product quantizations and q special functions theory are given among others the meromorphic continuation of the generalized basic hypergeometric series is found and a reproducing measure is constructed when the series is treated as a reproducing kernel

coherent orientations in symplectic field theory we study the coherent orientations of the moduli spaces of holomorphic curves in symplectic field theory generalizing a construction due to floer and hofer in particular we examine their behavior at multiple closed reeb orbits under change of the asymptotic direction the orientations are determined by a certain choice of orientation at each closed reeb orbit that is similar to the orientation of the unstable tangent spaces of critical points in finite dimensional morse theory

the boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations we investigate the boundary growth of positive superharmonic functions u on a bounded domain in mathbb r n n 3 satisfying the nonlinear elliptic inequality 0 le delta u le c delta omega x alpha u p quad rm in omega where c 0 0 and p 0 are constants and delta omega x is the distance from x to the boundary of the result is applied to show a harnack inequality for such superharmonic functions also we study the existence of positive solutions with singularity on the boundary of the nonlinear elliptic equation delta u vu f x u quad rm in omega where v and f are borel measurable functions conditioned by the generalized kato class