equivalence of q bosons using the exponential phase operator various forms of the q boson are explained and their hidden symmetry revealed by transformations using the exponential phase operator both the one component and the multicomponent q bosons are discussed as a byproduct we obtain a new boson algebra having a shifted vacuum structure and define a global operator u 1 gauge transformation

the q deformed boson realization of representations of quantum universal enveloping algebras for q a root of unity i the case of uqsl l the properties of q deformed boson operators with non generic q q is a root of unity are analysed by using the representation theory method and their finite dimensional representations are thereby obtained based on this discussion reducibilities and decompositions of q deformed boson realized representations of quantum universal enveloping algebra uqsl l are studied for non generic cases the explicit matrix elements of some indecomposable representations are obtained on the q deformed fock spaces necessary details are provided for uqsl 2 and uqsl 3 in particular the lusztig operator extension of uqsl 2 is discussed in an explicit form

a p q oscillator realization of two parameter quantum algebras it is noted that the study of a quantum algebra sup q 2 with two independent deformation parameters p q leads to a p q oscillator realization for it the analysis is extended to the p q analogues of su 1 1 osp 2 1 and the centreless virasoro algebra the standard single parameter q deformations are obtained in the limit p q

a two parameter deformation of the jaynes cummings model path integral representation a two parameter p q deformation of the jaynes cummings model is obtained using a recently developed p q deformed oscillator in the rotating wave approximation rwa the dynamical symmetry of the model is the quantum superalgebra up q 1 mod 1 the partition function of the model is obtained as a path integral over generalized perelomov coherent states corresponding to the quantum algebra up q 1 mod 1 the complete spectrum of the model is extracted for both the case when the coupling constants are grassman valued and when they are c number valued it is noted that a relaxation of the rwa extends the dynamical symmetry to the quantum superalgebra ospp q 2 mod 2

on the two q analogue logarithmic functions there is a simple multi sheet riemann surface associated with s inverse function for a principal sheet for can be defined however the topology of the riemann surface for changes each time q increases above the collision point of a pair of the turning points of there is also a power series representation for an infinite product representation for is used to obtain the ordinary natural logarithm and the values of the sum rules for the zeros of for where the values of the sum rules for the q trigonometric functions and are q deformations of the usual bernoulli numbers

u 1 1 coherent states and a path integral for the jaynes cummings model the path integral over the coherent states of the u 1 1 superalgebra is constructed it is applied to the jaynes cummings model whose dynamical group is u 1 1

quantum deformation of bose parastatistics a q deformation of the transformation of the chevalley basis to an odd basis of generators of the universal enveloping of the lie superalgebra g n osp 1 2n is presented it is shown that one thus obtains a reasonable quantum deformation of the algebra of n para bose oscillators

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

the q harmonic oscillator and an analogue of the charlier polynomials a model of a q harmonic oscillator based on q charlier polynomials of al salam and carlitz 1965 is discussed a simple explicit realization of q creation and q annihilation operators q coherent states and an analogue of the fourier transformation are found a connection of the kernel of this transform with biorthogonal rational functions is observed

complementarity of suq 3 and uq 2 and q boson realization of the suq 3 irreducible representations the complementarity relationship also termed duality that arises between the irreps of the su 3 and u 2 lie algebras when their direct sum su 3 u 2 is embedded into a larger u 6 algebra and single row irreps of the latter are considered is extended to the corresponding q algebras suq 3 and uq 2 it is demonstrated by explicitly constructing the unique q boson state that is simultaneously maximal in suq 3 and uq 2 for a given u q 2 weight in addition the relations between the suq 3 and uq 2 casimir operators resulting from their complementarity are explicitly found together with the q bargmann representation of q boson operators the complementarity relationship is then used to construct a gel fand tseitlin basis for arbitrary suq 3 irreps in terms of q boson operators

lyapunov exponent for the parabolic anderson model in rd abstract we consider the asymptotic almost sure behavior of the solution of the equation u t x u 0 x 2 0 t u s x d s 0 t u s x w x s where w x x r d is a field of brownian motions in fact we establish existence of the lyapunov exponent lim t 1 t log u t x we also show that c 1 1 3 c 2 1 5 as 0 under the assumption that the correlation function of the background field w x x r d is c for 1 2

the q deformed moszkowski model rpa modes the behaviour of the moszkowski model within the context of quantum algebras is studied the moszkowski hamiltonian is exactly diagonalized for various values of the deformation parameter of the involved quantum algebra the phase transition from a vibrational regime to a rotational regime is discussed in terms of the q deformation a coherent state for the ground state is introduced and the hartree fock energy and the rpa frequencies are compared with the exact values the meaning of the q deformation in this model is discussed

q analogues of the parabose and parafermi oscillators and representations of quantum algebras deformations of the one dimensional parabose oscillators are presented they are shown to provide fock representations of the quantum sp q 2 algebra and ospq 1 2 superalgebra the representations of suq 2 are used to define the q analogues of the parafermi oscillators

generalized commutators and deformation of strong coupling superconductivity we consider quantum deformation of a strong coupling superconductivity model based on creation annihilation operators which satisfy generalized commutator relations it is found that the nature of the superconducting phase transition can be changed from the usual second order to a first order transition if the deformation parameter exceeds a certain critical value metastable normal and superconducting states can exist when the transition is of the first order

extension of q deformed analysis and q deformed models of classical mechanics the author presents the generalization of the q derivative and investigates its properties the formula of conjugation for scalar products with respect to the lebesgue and discrete measures is obtained this formalism is applied to classical mechanical systems depending on functional derivatives he derives the deformed euler lagrange equations and deformed poisson bracket by an assumption that it determines the functional evolution of the systems it appears that in such models a hamiltonian analogue is no longer conserved in time for a few examples the explicit forms of constants of evolution are given

a q deformed oscillator system with quantum group slq l symmetry a q deformed oscillator system with quantum group slq l symmetry is constructed in terms of the q deformed boson operators the energy level degeneracy associated with the irreducible representation of slq l is analysed especially for the case that q is a root of unity

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

one dimensional partially asymmetric simple exclusion process with open boundaries orthogonal polynomials approach the stationary state of the partially asymmetric simple exclusion process with open boundaries is reconsidered the so called matrix product ansatz is employed this enables us to construct the stationary state in the form of the product of the matrices d and e noticing the fact that the matrix c d e for the model is related to certain q orthogonal polynomials the model is analysed for a wide range of parameters the current and the correlation length are evaluated in the thermodynamic limit it turns out that the phase diagram for the correlation length is richer than that for the totally asymmetric case

the q deformed boson realization of representation of quantum universal enveloping algebras for q a root of unity ii the subalgebra chain of uq cl for pt i see ibid vol 24 p 3265 1991 applying the method of the q deformed boson realization for the quantum group to the case where q is a root of unity the authors establish a set of standard basis for the representation space of the quantum universal enveloping algebra uq cl associated with a typical subalgebra chain of uq cl on this basis they systematically obtain irreducible and indecomposable representations of uq cl and its subalgebras they discuss uq c2 in detail

representations of quantum superalgebra uq gl 2 1 in a coherent state basis and generalization the coherent state method has proved to be useful in quantum physics and mathematics this method more precisely the vector coherent state method has been used by some authors to construct representations of superalgebras but almost to our knowledge it has not yet been extended to quantum superalgebras except uq osp 1 2 one of the smallest quantum superalgebras in this article the method is applied to a bigger quantum superalgebra namely uq gl 2 1 in constructing q boson fermion realizations and finite dimensional representations which when irreducible are classified into typical and nontypical representations this construction leads to a more general class of q boson fermion realizations and finite dimensional representations of uq gl 2 1 and thus at q 1 of gl 2 1 both gl 2 1 and uq gl 2 1 have found different physics applications therefore it is meaningful to construct their representations

the poisson bracket for q deformed systems it is shown that the quantum system of q deformed oscillators with the uq n group symmetry can be obtained by the canonical quantization of a classical system with a modified poisson bracket the modification of the poisson bracket is connected with a non canonical transformation of phase space variables in this approach the deformation parameter turns out to be a function of the planck constant and some dimensional parameters characterizing the classical system

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

generating function for clebsch gordan coefficients of the suq 2 quantum algebra some methods have been developed to calculate the suq 2 clebsch gordan coefficients cgc here the authors develop a method based on the calculation of clebsch gordan generating functions through the use of quantum algebraic coherent states calculating the suq 2 cgc by means of this generating function is an easy and straightforward task

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

realizability of a model in infinite statistics following greenberg and others we study a space with a collection of operatorsa k satisfying the q mutator relations a l a k a l k l corresponding forq 1 to classical bose and fermi statistics we show that then n matrixa n q representing the scalar products ofn particle states is positive definite for alln ifq lies between 1 and 1 so that the commutator relations have a hilbert space representation in this case this has also been proved by fivel and by bozejko and speicher we also give an explicit factorization ofa n q as a product of matrices of the form 1 q jt 1 with 1 j n andt a permutation matrix in particular a n q is singular if and only ifq m 1 for some integerm of the formk 2 k 2 k n

real de sitter and conformal algebras in terms of q oscillators we express the cartan weyl basis of all quantum real d 4 de sitter and d 4 conformal algebras in terms of q deformed oscillators

comment on the q deformed fermionic oscillator the q deformed fermionic oscillator is identified with the usual fermionic oscillator by virtue of a simple transformation so the realization of quantum algebra suq 2 can also be constructed from the usual fermionic oscillator

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

nonlinear lie algebra and ladder operators for orbital angular momentum based on the nonlinear lie algebra a new framework is put forward to treat the problem of angular momentum from this framework the ladder operators for the orbital angular momentum can be solved as simultaneous eigenvectors of the lie derivations ad l2 and ad l2 in an operator space the ladder operators for solvable central potentials can also be found in a united way

exponential decay rate of survival probability in a disastrous random environment we study the exponential decay rate of the survival probability up to time t 0 of a random walker moving in zopf d in a temporally and spatially fluctuating random environment when the random walker has a speed parameter 0 we investigate the influence of on the exponential decay rate d in particular we prove that for any fixed d 1 d behaves like as log as 0

incomplete 2d hermite polynomials properties and applications incomplete forms of two variable two index hermite polynomials are introduced their link with laguerre polynomials is discussed and it is shown that they are a useful tool to study quantum mechanical harmonic oscillator entangled states the possibility of developing the theory of complete 2d hermite polynomials from the point of view of the incomplete forms is analyzed too the orthogonality properties of the associated harmonic oscillator functions are finally discussed

multi parametric deformed heisenberg algebras a route to complexity we introduce a generalization of the heisenberg algebra which is written in terms of a functional of one generator of the algebra f j0 that can be any analytical function when f is linear with slope we show that the algebra in this case corresponds to q oscillators for q2 tan the case where f is a polynomial of order n in j0 corresponds to an n parameter deformed heisenberg algebra the representations of the algebra when f is any analytical function are shown to be obtained through the study of the stability of the fixed points of f and their composed functions the case when f is a quadratic polynomial in j0 the simplest nonlinear scheme which is able to create chaotic behaviour is analysed in detail and special regions in the parameter space give representations that cannot be continuously deformed to representations of heisenberg algebra

generalized q oscillators and their hopf structures we study the relationships among the various forms of the q oscillator algebra and consider the conditions under which it supports a hopf structure we also present a generalization of this algebra together with its corresponding hopf structure its multimode extensions are also considered

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

three dimensional quadratic algebras some realizations and representations four classes of three dimensional quadratic algebra of the type q0 q q q q aq02 bq0 c where a b c are constants or central elements of the algebra are constructed using a generalization of the well known two mode bosonic realizations of su 2 and su 1 1 the resulting matrix representations and single variable differential operator realizations are obtained some remarks on the mathematical and physical relevance of such algebras are given

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

evolution operators for quantum chains discrete time evolution operators in integrable quantum lattice models are sometimes more fundamental objects than hamiltonians in this paper we study an evolution operator for the one dimensional integrable q deformed bose gas with xxz type impurities and find its spectrum evolution operators give a new interpretation of known integrable systems for instance our system describes apparently a simplest laser with a clear resonance peak in the spectrum

brst cohomology for uq sl 2 representations a nilpotent brst operator qq is constructed for the q deformation of the universal enveloping algebra of sl 2 an associated operator rq such that qq rq cq is introduced where cq is the q deformed casimir invariant cq fe 1 2h q 1 2h 1 q the related operator qq conjugate to qq is also considered the resulting brst cohomology of uq sl 2 representations is discussed

summation formulae for gould hopper generalized hermite polynomials in this article we derive summation formulae for the gould hopper generalized hermite polynomials g n m x y further we derive the summation formulae for polynomials related to g n m x y as applications furthermore we establish a summation formula for m variable generalized hermite polynomials h n 1 2 m x 1 x 2 x m

unitary representations of non compact supergroups we give a general theory for the construction of oscillator like unitary irreducible representations uirs of non compact supergroups in a super fock space this construction applies to all non compact supergroupsg whose coset spaceg k with respect to their maximal compact subsupergroupk is hermitean supersymmetric we illustrate our method with the example of su m p n q by giving its oscillator like uirs in a particle state basis as well as supercoherent state basis the same class of uirs can also be realized over the super hilbert spaces of holomorphic functions of az variable labelling the coherent states

the distribution of zeros of general q polynomials a general system of q orthogonal polynomials is defined by means of its three term recurrence relation this system encompasses many of the known families of q polynomials among them the q analogue of the classical orthogonal polynomials the asymptotic density of zeros of the system is shown to be a simple and compact expression of the parameters which characterize the asymptotic behaviour of the coefficients of the recurrence relation this result is applied to specific classes of polynomials known by the names q hahn q kravchuk q racah q askey and wilson al salam carlitz and the celebrated little and big q jacobi

approach of spherical harmonics to the representation of the deformed su 1 1 algebra the m shifting generators of su 2 algebra together with a pair of l shifting ladder symmetry operators have been used in the space of all spherical harmonics ylm in order to introduce a new set of operators expressing the transitions between them it is shown that the space of spherical harmonics whose l 2m or l 2m is given presents negative and positive irreducible representations of a deformed su 1 1 algebra respectively these internal symmetries also suggest new algebraic methods to construct the spherical harmonics in the framework of the spectrum generating algebras

a q analog of the quantum central limit theorem for suq 2 a q analog of the central limit theorem for suq 2 q 0 is studied it is shown that the limits of the moments and q exponential generating functions for coherent states give for q 1 0 q 1 the harmonic q oscillator q 1 oscillator introduced by biedenharn and macfarlane

generalized q fermion oscillators and q coherent states the algebra of q fermion operators developed earlier by two of the present authors is re examined it is shown that these operators represent particles that are distinct from usual spacetime fermions except in the limit q 1 it is shown that it is possible to introduce generalized q oscillators defined for infinity q or 1 in the range infinity q 0 these coincide with the q boson operators and for 0 q or 1 they coincide with q fermions the ordinary bosons and fermions may be identified with the limits q 1 and 1 respectively generalized q fermion coherent states are constructed by utilizing a nonlinear shift automorphism of the algebra of q fermion operators these are compared with the coherent states defined as eigenstates of annihilation operator matrix elements of the shift operator in the fock space basis are evaluated

suq h cross to 0 2 and suq h cross 2 the classical and quantum q deformations of the su 2 algebra ii the hopf algebra the yang baxter equation and multi deformed algebraic structures for pt i see ibid vol 23 p 4185 1990 the suq 2 algebra is realized by means of both the poisson brackets in classical mechanics and commutators in quantum mechanics in a system with q deformed oscillators of two different types the structures of the hopf algebra and the quantum yang baxter equation are also discussed on a quantum level a set of j representations of the quantum algebra suq 2 is constructed based on the type ii q oscillators multi deformations of the oscillators of the two types and multi deformed algebras expressed in poisson brackets as well as in lie brackets are proposed

a simple difference realization of the heisenberg q algebra a realization of the heisenberg q algebra whose generators are first order difference operators on the full real line is discussed herein the eigenfunctions of the corresponding q oscillator hamiltonian are given explicitly in terms of the q 1 hermite polynomials the nonuniqueness of the measure for these q oscillator states is also studied

a study of representations of the algebra of functions on the quantum group glq n a q analogue of the root system is constructed for this algebra which is similar to the root system of classical lie algebras it is then used to construct in detail a class of representations of the algebra of functions on the quantum group glq n and a q boson realization of the generators of glq n is given i also construct infinite dimensional hilbert space representations of this algebra the main result of this paper is stated in proposition 8

verma module of the quantum group gl n q and its q boson and heisenberg weyl relation realizations the structure and verma module of the quantum matrix element algebra a n q of the quantum group gl n q are studied the method used here is similar to that for studying the structure and verma module of semisimple lie algebras the concept of the cartan subalgebra raising and lowering operators and their pairs is defined the q boson and the corresponding heisenberg weyl relation realizations are generally studied and the cyclic representations of a n q are obtained the explicit examples a 2 q and a 3 q are discussed in detail

algebraic structure of parabose fock space i the green s ansatz revisited a description of the fock space of a parabose oscillator of order p based on commutation relations bilinear in creation and annihilation operators rather than trilinear ones is developed a new statement of the well known green s ansatz is introduced to help understand the significance of the bilinear commutation relations it is also applied to show how all parabose oscillators of even order respectively odd order can be obtained as irreducible constituents of the fock space associated with a green s ansatz of two three terms analogs of the form in which the parabose green s ansatz is presented are provided for one parafermi oscillator and for systems of many parabose and parafermi oscillators of the same order methods used in the paper allow various new results for q deformed parabose oscillators to be derived

path integral formalism for suq 2 coherent states a path integral formulation in the representation of coherent states for the quantum group suq 2 is introduced an expression for the transition amplitude connecting two suq 2 coherent states is constructed and the corresponding canonical equations of motion are also derived

a deformed oscillator with coulomb energy spectrum a deformed oscillator with eigenvalues equal to the eigenvalues of the schrodinger equation with the coulomb potential is constructed the deformed oscillator algebra has a polynomial realization and the associate deformed operations of integration and differentiation are studied

implicit summation formulae for hermite and related polynomials in this article we derive some implicit summation formulae for hermite and related polynomials by using different analytical means on their respective generating functions

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

a new completeness relation in the q deformed two mode fock space in two mode fock space a new completeness relation for the q analogue of the single complex variable state is proved by virtue of some q deformed combinatorial identities which are also derived here for the first time

on position and momentum operators in the q oscillator algebra the aim of this paper is to study the position and momentum operators in q deformed oscillator algebras the natural form of the position operator is xp qpn a a qpn where p is a real number this operator is an operator representable by a jacobi matrix using the theory of jacobi matrices the theory of classical moment problem and the theory of basic hypergeometric functions it is shown that depending on values of q and p xp can be unbounded symmetric operator which has the deficiency indices 1 1 and hence is not self adjoint but has self adjoint extensions bounded self adjoint operator with continuous simple spectrum or self adjoint operator of trace class therefore with discrete spectrum with zero as the point of accumulation of eigenvalues the connection of the q deformed heisenberg relation px qxp 1 for the position and momentum operators with a q deformation of the quantum harmonic oscillator is also considered

q oscillators realization of fq 4 and gq 3 a realization of the quantum superalgebras corresponding to f 4 and g 3 is given in terms of q deformed bosonic and fermionic oscillators

unitarizable representations of the deformed para bose superalgebra uq osp 1 2 at roots of 1 the unitarizable irreps of the deformed para bose superalgebra pbq which is isomorphic to uq osp 1 2 are classified at q being root of 1 new finite dimensional irreps of uq osp 1 2 are found explicit expressions for the matrix elements are written down

extension of the q deformed oscillator and two parameter coherent states the suq 2 algebra is extended by introducing additional raising and lowering operators and constructing their coherent states this new algebra of coherent states and the commutation relations between the extended operators are investigated and a resolution of unity is proposed

the q boson realization of parametrized cyclic representations of quantum algebras at qp 1 a parametrized cyclic representation of the q deformed heisenberg weyl algebras is constructed the cyclic representations of any quantum algebras are studied in terms of their q boson realizations a general method to construct the q boson realizations of quantum algebras from their verma representations is proposed two explicit examples sl 2 q and sl 3 q are studied in detail some cyclic representations of quantum algebras sl n q and uqcn are presented by using their q boson realizations

on large deviations for the parabolic anderson model the focus of this article is on the different behavior of large deviations of random functionals associated with the parabolic anderson model above the mean versus large deviations below the mean the functionals we treat are the solution u x t to the spatially discrete parabolic anderson model and a functional an which is used in analyzing the a s lyapunov exponent for u x t both satisfy a law of large numbers with lim t to infty frac 1 t log u x t lambda kappa and lim n to infty frac a n n alpha we then think of n and t as being the mean of the respective quantities an and log u t x typically the large deviations for such functionals exhibits a strong asymmetry large deviations above the mean take on a different order of magnitude from large deviations below the mean we develop robust techniques to quantify and explain the differences

the q boson realizations of the quantum group uq sl n 1 c the authors give an explicit expression of the recurrency formulae of canonical realization for quantum enveloping algebras uq sl n 1 c in these formulae the generators of the algebra uq sl n 1 c are expressed by means of n canonical q boson pairs one auxiliary representation of the algebra uq gl n c

jacobson generators fock representations and statistics of sl n 1 the properties of a statistics related to the class a of simple lie algebras t d palev preprint jinr e17 10550 1977 hep th 9705032 are further investigated the description of each sl n 1 is carried out via generators a1 an which we call jacobson generators with respect to these generators the definition of a fock space of sl n 1 is given it is proved that the fock spaces wp p n are the simple symmetric finite dimensional modules of sl n 1 the pauli principle of the underlying statistics is formulated within each wp operators b p i ai p i 1 n called quasi bose creation and annihilation operators caos are defined then limp b p i bi where bi are ordinary bose caos therefore a statistics appears as an approximation of bose statistics with caos acting in finite dimensional state spaces we indicate that the p 1 quasi bose operators b 1 1 b 1 n are natural operators for the description of hard core bose models and of the related heisenberg spin models we argue th

a q analogue of the supersymmetric oscillator and its q superalgebra a q analogue of the supersymmetric oscillator is constructed out of q boson and q fermion creation and annihilation operators for a fixed nb nf 2n where nb and nf are q boson and q fermion occupation numbers the irreducible representations of the q superalgebra generated by the q oscillator are 2n 1 dimensional particular cases when q is a root of or 1 are discussed a realization of the quantum group suq 2 is obtained using a pair of 1 fermion creation and annihilation operators

on quantum deformations of the simplest lie parasuperalgebra quantum deformations of the simplest lie parasuperalgebra psqm 2 are constructed by taking advantage of the well known representations of suq 2 c as a particular result the authors show that the fundamental witten superalgebra sqm 2 cannot be deformed

integrable representations of the ultra commutation relations we consider representations of the unital complex algebra generated by the identity and two elements and with and one relation the ultra commutation relations ucr in general we do not impose any commutation relation between and this is a very general scheme encompassing many important physical examples inter alia the ccr car q deformed bosons and fermions the representations of interest in physics have a diagonal number operator whose spectrum is contained in the positive integers together with some other technical conditions our principal result is that every representation in this class is completely determined up to unitary equivalence by the sequence of numbers n 1 n 1 n 2 for n 0 with 0 0 here n is the normalized eigenvector of corresponding to the eigenvalue n if the dimension of that eigenspace is 1 if the carrier hilbert space is infinite dimensional this representation is irreducible if and only if n 0 for n 1 finall

hopf superalgebra contractions and r matrix for fermions different contractions are applied to the quantum superalgebra ospq 1 mod 2 in the first of them the graded analogue s hq 1 of the one dimensional heisenberg quantum algebra is obtained and its r matrix explicitly calculated in the second contraction a z2 graded version of the q oscillator is proposed and finally the supersymmetric two dimensional euclidean quantum algebra s eq 2 is found

new relations for two dimensional hermite polynomials the effective formulas reducing the two dimensional hermite polynomials to the classical one dimensional orthogonal polynomials by jacobi gegenbauer legendre laguerre and hermite are given new one parameter generating functions for the diagonal multidimensional hermite polynomials are derived the factorial moments and cumulants of the distribution functions related to the hermite polynomials of two variables with equal indices are expressed in terms of the legendre and chebyshev polynomials asymptotical formulas for the two dimensional polynomials with large values of indices and zero arguments are found the applications to the squeezed one mode states and to the time dependent quantum harmonic oscillator are considered

generalized heisenberg algebra application to the harmonic oscillator the deformed poisson algebra recently introduced to investigate integrable systems 2003 j phys a math gen 36 12181 203 2005 j math phys 46 042702 is used to perform the transition from the phase space of classical observables functions depending on positions and momentums to the hilbert space of physically well defined hermitian operators a hamiltonian operator for the harmonic oscillator system is constructed and the eigenvalue problem is solved the generalization to an n dimensional space shows that such an algebra does not break the rotational symmetry

on a q generalization of circular and hyperbolic functions a generalization of the circular and hyperbolic functions is proposed based on the tsallis statistics and on a consistent generalization of the euler formula some properties of the presently proposed q trigonometry are then investigated the generalized functions are exact solutions of a nonlinear oscillator original circular and hyperbolic relations are recovered as the limiting case

an algebraic q deformed form for shape invariant systems a quantum deformed theory applicable to all shape invariant bound state systems is introduced by defining q deformed ladder operators we show that these new ladder operators satisfy new q deformed commutation relations in this context we construct an alternative q deformed model that preserves the shape invariance property presented by the primary system q deformed generalizations of morse scarf and coulomb potentials are given as examples

the q supercoherent states of the q deformed su 2 superalgebra the q supercoherent states associated with the q deformed su 2 superalgebra are constructed explicitly and their properties are investigated in detail the completeness relation of the q supercoherent states is proved by the use of the q integration

alternative hamiltonian quantization the authors take a standard hamiltonian quadratic in a set of creation and annihilation operators and demonstrate an alternative quantization by quommutators i e a set of rules for transposition of these operators different from the those of the usual canonical commutation relations which yields a new spectrum of a type reminiscent of logarithmic trajectories

hermite and laguerre 2d polynomials abstract we define hermite 2d polynomials hm n u x y and laguerre 2d polynomials l m n u z z as functions of two variables with an arbitrary 2d matrix u as parameter and discuss their properties and their explicit representation recursion relations and generating functions for these polynomials are derived the advantage of the introduced hermite and laguerre 2d polynomials in comparison to the related usual two variable hermite polynomials is that they satisfy orthogonality relations in a direct way whereas for the purpose of orthonormalization of the last one has to introduce two different kinds of such polynomials which are biorthogonal to each other