SU(2) q in a Hilbert space of analytic functions

The algebraSU(2) _q is realized in a Hilbert spaceH _q ² of analytic functions; the starting point is the differential realization of operators that satisfyq-algebra in a Hilbert spaceH _q. The Weyl realization ofSU(2) _q is constructed exhibiting the reproducing kernel and the principal vectors; the noncommutativity of the matrix elements of a 2×2 linear representation ofSU(2) _q is obtained as consistency conditions for couplingj1=j2=1/2 toj=0, 1; the derivation of Clebsch-Gordan coefficients is sketched and theq-generalization of the rotation matrices is included. The unitary correspondence ofH _q with a Hilbert space of complex functions of a real variable is also studied. The study presented in this paper follows Bargmann's formalism for the rotation group as closely as possible.     

Big q-Jacobi polynomials,q-Hahn polynomials,and a family of quantum 3-spheres

The big q-Jacobi polynomials and the q-Hahn polynomials are realized as spherical functions on a new quantum SU _q (2)-space which can be regarded as the total space of a family of quantum 3-spheres.     

Freeq-Schrödinger equation from homogeneous spaces of the 2-dim Euclidean quantum group

After a preliminary review of the definition and the general properties of the homogeneous spaces of quantum groups, the quantum hyperboloidqH and the quantum planeqP are determined as homogeneous spaces ofF _q (E(2)). The canonical action ofE _q (2) is used to define a naturalq-analog of the free Schrödinger equation, that is studied in the momentum and angular momentum bases. In the first case the eigenfunctions are factorized in terms of products of twoq-exponentials. In the second case we determine the eigenstates of the unitary representation, which, in theqP case, are given in terms of Hahn-Exton functions. Introducing the universalT-matrix forE _q (2) we prove that the Hahn-Exton as well as Jacksonq-Bessel functions are also obtained as matrix elements ofT, thus giving the correct extension to quantum groups of well known methods in harmonic analysis.     

An extension of the Borel-Weil construction to the quantum groupU q (n)

The Borel-Weil (BW) construction for unitary irreps of a compact Lie group is extended to a construction of all unitary irreps of the quantum groupU _q(n). Thisq-BW construction uses a recursion procedure forU _q(n) in which the fiber of the bundle carries an irrep ofU _q(n–1)×U _q(1) with sections that are holomorphic functions in the homogeneous spaceU _q(n)/U _q(n–1)×U _q(1). Explicit results are obtained for theU _q(n) irreps and for the related isomorphism of quantum group algebras.Supported in part by the National Science Foundation, No. PHY-9008007     

Unitary representations of the quantum group SU q (1, 1): II — Matrix elements of unitary representations and the basic hypergoemetric functions

Some series of unitary representations of the quantum group SU _q (1, 1) are introduced. Their matrix elements are expressed in terms of the basic hypergeometric functions. Operator realization of the coordinate elements of SU _q (1, 1) and aq-analogue of some classical identities are discussed.     

Λ(1405) as a multiquark state

In the QCD sum rule approach we predict the Λ (1405) mass by choosing the π⁰Σ⁰ multiquark interpolating field. It is found that the mass is about 1.419 GeV from Π₁ (q ²) sum rule which is more reliable than Π_q (q ²) sum rule, where Π_q (q ²) and Π₁ (q ²) are two invariant functions of the correlator Π (q ²). We also present the sum rules for the K ⁺ p and the π⁺Σ⁺ multiquark states, and compare to those for the π⁰Σ⁰ multiquark state. The mass of the Λ (1600) can be also reproduced in our approach. Received: 11 November 1997 / Revised version: 28 April 1998     

Trapezoidal (p,q)-Integral Inequalities Related to (η1,η2)-convex Functions with Applications

In this paper, the authors introduce the (p,q)-trapezoidal integral inequalities, which are the (p,q)-analogues of the recently introduced q-trapezoidal integral inequalities. We derive a new (p,q)-integral identity for twice (p,q)-differentiable function. Utilizing this as an auxiliary result, we establish several new (p,q)-trapezoidal type integral inequalities for the function whose absolute value of twice (p,q)-derivative is (η₁,η₂)-convex functions. Some special means of real numbers are also given. At the end, we give brief conclusion. It is expected that this method which is very useful, accurate, and versatile will open a new venue for the real-world phenomena of special relativity and quantum theory.

     

Dynamical implications of gluonic excitations in meson-meson systems

We study meson-meson interactions using an extended q ² [`(q)]<sup>2</sup> (g)\bar q^2 (g) basis that allows calculating coupling of an ordinary meson-meson system to a hybrid-hybrid one. We use a potential model matrix in this extended basis which at quark level is known to provide a good fit to numerical simulations of a q <sup>2</sup> [`(q)]²\bar q^2 system in pure gluonic theory for static quarks in a selection of geometries. We use a combination of resonating group method formalism and Born approximation to include the quark motion using wave functions of a q[`(q)]q\bar q potential within a cluster. This potential is taken to be quadratic for ground states and has an additional smeared $\frac{1} {r}$\frac{1} {r} (Gaussian) for the matrix elements between hybrid mesons. For the parameters of this potential, we use values chosen to 1) minimize the error resulting from our use of a quadratic potential and 2) best fit the lattice data for differences of Σ _g and Π _u configurations of the gluonic field between a quark and an antiquark. At the quark (static) level, including the gluonic excitations, was noted to partially replace the need for introducing many-body terms in a multiquark potential. We study how successful such a replacement is at the (dynamical) hadronic level of relevance to actual hard experiments. Thus we study the effects of both gluonic excitations and many-body terms on mesonic transition amplitudes and the energy shifts resulting from the second-order perturbation theory (i.e. from the respective hadron loops). The study suggests introducing both energy and orbital excitations in wave functions of scalar mesons that are modelled as meson-meson molecules or are supposed to have a meson-meson component in their wave functions.     

Thermodynamics of two-parameter quantum group Bose and Fermi gases

The high and low temperature thermodynamical properties of the two-parameter deformed quantum group Bose and Fermi gases with SU _p/q (2) symmetry are studied. Starting with a SU _p/q (2)-invariant bosonic as well as fermionic Hamiltonian, several thermodynamical functions of the system such as the average number of particles, internal energy and equation of state are derived. The effects of two real independent deformation parameters p and q on the properties of the systems are discussed. Particular emphasis is given to a discussion of the Bose-Einstein condensation phenomenon for the two-parameter deformed quantum group Bose gas. The results are also compared with earlier undeformed and one-parameter deformed versions of Bose and Fermi gas models. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Realization ofU q (so(N)) within the differential algebra onR q N

We realize the Hopf algebraU _q–1 (so(N)) as an algebra of differential operators on the quantum Euclidean spaceR _q ^N . The generators are suitableq-deformed analogs of the angular momentum components on ordinaryR ^N . The algebra Fun(R _q ^N ) of functions onR _q ^N splits into a direct sum of irreducible vector representations ofU _q–1 (so(N)); the latter are explicitly constructed as highest weight representations.     

Quantized Knizhnik-Zamolodchikov equations,quantum Yang-Baxter equation,and difference equations forq-hypergeometric functions

Thes₂ quantized Knizhnik-Zamolodchikov equations are solved inq-hypergeometric functions. New difference equations are derived for generalq-hypergeometric functions. The equations are given in terms of quantum Yang-Baxter matrices and have the form similar to quantum Knizhnik-Zamolodchikov equations for quantum affine algebras introduced by Frenkel and Reshetikhin.This work was supported by NSF grant DMS-9203929.     

Left regular representation and contraction of sl q (2) to e q (2)

The left regular representation of the quantum algebras sl _q (2) and e _q (2) are discussed and shown to be related by contraction. The reducibility is studied andq-difference intertwining operators are constructed.     

Representations of the cyclically symmetric q-deformed algebra U q(so3)

An algebra homomorphism from the nonstandard q-deformed (cyclically symmetric) algebra U _q(so₃) to the extension Û _q(sl₂) of the Hopf algebra U _q(sl₂) is constructed. Not all irreducible representations (IR) of U _q(sl₂) can be extended to representations of Û _q(sl₂). Composing the homomorphism with irreducible representations of Û _q(sl₂) we obtain representations of U _q(so₃). Not all of these representations of U _q(so₃) are irreducible. Reducible representations of U _q(so₃) are decomposed into irreducible components. In this way we obtain all IR of U _q(so₃) when q is not a root of unity. A part of these representations turn into IR of the Lie algebra so₃ when q 1.     

On the invariant distance and Green function for SU q(2)

Following the introduction of the invariant distance on the non-commutative C-algebra of the quantum group SU _q(2) the Green function on the q-Podler's sphere M _q = SU _q(2)/U(1) is derived. Possible applications are briefly discussed.     

On the quantum group and quantum algebra approach toq-special functions

Two interpretations ofq-special functions based on quantum groups and algebras have been presented in the literature. The connection between these approaches is explained using as an example the case whereU _q (sl(2)) is the basic structure.Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada.     

Electron correlation effects in N2 and CO studied by X-ray scattering and CISD calculations

Total X-ray scattering intensities σ_ee(q) for N₂ and CO have been measured as a function of momentum transfer using the energy dispersive method. Novel procedures to extract accurate σ_ee(q), which eliminate effects of polarization, inelastic scattering, anomalous dispersion, and molecular vibration, have been proposed. A simplified theoretical treatment based on configuration interaction singles and doubles (CISD) calculations has been suggested. This procedure makes it possible to apply combined theoretical and experimental X-ray scattering studies to larger molecules. The inclusion of f and g functions is crucial, and the σ_ee(q) calculated with the cc-pVQZ[5s4p3d2f1g] basis set almost reproduces the data based on more elaborate MR-CISD calculations within the experimentally most relevant region of up to q ? 3 au. In contrast to experimental electron scattering data, the X-ray scattering intensities agree well with the computed results.     

A Hopf Bundle Over a Quantum Four-Sphere from the Symplectic Group

We construct a quantum version of the SU(2) Hopf bundle S⁷→S⁴. The quantum sphere S⁷_q arises from the symplectic group Sp_q(2) and a quantum 4-sphere S⁴_q is obtained via a suitable self-adjoint idempotent p whose entries generate the algebra A(S⁴_q) of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere S⁴. We compute the fundamental K-homology class of S⁴_q and pair it with the class of p in the K-theory getting the value −1 for the topological charge. There is a right coaction of SU_q(2) on S⁷_q such that the algebra A(S⁷_q) is a non-trivial quantum principal bundle over A(S⁴_q) with structure quantum group A(SU_q(2)).     

Asymptotic Approximations in Quantum Calculus

Abstract

This paper aims to study the asymptotic approximation of some functions defined by the q-Jackson integrals, for a fix q ∈]0, 1[. For this purpose, we shall attempt to extend the classical methods by giving their q-analogues. In particular, a q-analogue of the Watson’s lemma is discussed and new asymptotic expansions of the q?j _α Bessel function and of the q-complementary error function are established.     

Integrable Equations of the Form q t =L 1(x,t,q,q x,q xx )q xxx +L 2(x,t,q,q x,q xx )

Integrable equations of the form q _t =L ₁(x,t,q,q _x ,q _xx )q _xxx +L ₂(x,t,q,q _x ,q _xx ) are considered using linearization. A new type of integrable equations which are the generalization of the integrable equations of Fokas and Ibragimov and Shabat are given.     

Valence bond analysis of the bonding in transition metal compounds: the RuNO group in nitrosyl complexes

The ab initio and semi-empirical configuration interaction wave functions of ruthenium complexes [RuL₅(XY) ^q (L &; = NH₃, Cl^?, CN^?, XY &;= N₂, CO) are presented in the form of linear combinations of the valence bond (VB) structures, each structure being referred to some covalent or ionic model of the bonding in the M-XY group. The results of this VB analysis showed that [RuL₅(NO)] ^q complexes can be described as compounds of Ru(III) and neutral NO⁰ with a covalent π-bond in addition to the usual coordination bond.