QCD analysis of structure functions in terms of Jacobi polynomials

A new method of QCD-analysis of singlet and nonsinglet structure functions based on their expansion in orthogonal Jacobi polynomials is proposed. An accuracy of the method is studied and its application is demonstrated using the structure functionF ₂(x, Q ²) obtained by the EMC Collaboration from measurements with an iron target.     

QCD analysis of the <Emphasis Type="Italic">F</Emphasis><Subscript>3</Subscript> structure function based on inverse Mellin transform in analytic perturbation theory

A QCD analysis of combined experimental data on the F ₃ structure function is performed using the inverse Mellin transform in the framework of analytical perturbation theory. Within this approach, the form of the F ₃ structure function, the value of the QCD scale parameter Λ, and the x dependence of the higher twist contribution are determined. The accuracy of the method based on Jacobi polynomials is estimated.     

F L structure function and heavy quark contributions

In this paper we obtain the heavy-quark contribution to the longitudinal structure functions F _L (x, Q ²). Since F _L structure functions contains rather large heavy flavor contributions in the small x region, we need to use the massive operator matrix elements, which contribute to the heavy flavor Wilson coefficients in unpolarized deeply inelastic scattering in the region Q ²?>?>?m ². The method of QCD analysis, based on the Jacobi polynomials method, is also described. Our results for longitudinal structure function are in good agreement with the available experimental data.     

Application of jacobi polynomial methods to the one-speed transport equation

The transport equations associated with radiation damage studies are often solved using expansions in Legendre polynomials. The radiation damage distribution functions which satisfy these equations may be sharply peaked in the forward direction, while the Legendre polynomials, as a set, are isotropic. This situation requires the use of many terms in the Legendre expansion in order to adequately represent the distribution functions. The Jacobi polynomials, on the other hand, can have strong peaking built into their associated weight function. To test the usefulness of the Jacobi polynomials we use them to solve the simple, one-speed, neutron transport equation. The results are then compared to the exact theory and to the results of applying Legendre methods to the same problem. This sample calculation demonstrates the advantage of the Jacobi polynomials in strongly non-isotropic situations.     

The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra

The superintegrability, wavefunctions and overlap coefficients of the Dunkl oscillator model in the plane were considered in the first part. Here finite-dimensional representations of the symmetry algebra of the system, called the Schwinger–Dunkl algebra sd(2), are investigated. The algebra sd(2) has six generators, including two involutions and a central element, and can be seen as a deformation of the Lie algebra \({\mathfrak{u}(2)}\) . Two of the symmetry generators, J ₃ and J ₂, are respectively associated to the separation of variables in Cartesian and polar coordinates. Using the parabosonic creation/annihilation operators, two bases for the representations of sd(2), the Cartesian and circular bases, are constructed. In the Cartesian basis, the operator J ₃ is diagonal and the operator J ₂ acts in a tridiagonal fashion. In the circular basis, the operator J ₂ is block upper-triangular with all blocks 2 × 2 and the operator J ₃ acts in a tridiagonal fashion. The expansion coefficients between the two bases are given by the Krawtchouk polynomials. In the general case, the eigenvectors of J ₂ in the circular basis are generated by the Heun polynomials, and their components are expressed in terms of the para-Krawtchouk polynomials. In the fully isotropic case, the eigenvectors of J ₂ are generated by little ?1 Jacobi or ordinary Jacobi polynomials. The basis in which the operator J ₂ is diagonal is considered. In this basis, the defining relations of the Schwinger–Dunkl algebra imply that J ₃ acts in a block tridiagonal fashion with all blocks 2 × 2. The matrix elements of J ₃ in this basis are given explicitly.     

On methods of analyzing scaling violation in deep inelastic scattering

We discuss relative merits and shortcomings of various methods used in analyses of deep inelastic scattering data within the framework of leading twist QCD perturbation expansion. We advocate the use of Jacobi polynomials method as by far the simples, fastest and simultaneously very accurate way of analysing theQ ²-evolution of nucleon structure functions. Detailed comparison with the numerical solution of the corresponding evolution equation to the next-to-leading order is presented.     

Exceptional orthogonal polynomials and exactly solvable potentials in position dependent mass Schrödinger Hamiltonians

Some exactly solvable potentials in the position dependent mass background are generated whose bound states are given in terms of Laguerre- or Jacobi-type X₁ exceptional orthogonal polynomials. These potentials are shown to be shape invariant and isospectral to the potentials whose bound state solutions involve classical Laguerre or Jacobi polynomials.     

Dispersion Estimates for the Discrete Laguerre Operator

We derive an explicit expression for the kernel of the evolution group \({\exp(-\mathrm{i} t H_0)}\) of the discrete Laguerre operator H₀ (i.e., the Jacobi operator associated with the Laguerre polynomials) in terms of Jacobi polynomials. Based on this expression, we show that the norm of the evolution group acting from \({\ell^1}\) to \({\ell^\infty}\) is given by \({(1+t^2)^{-1/2}}\).     

Parton distribution functions up to NNLO

We calculate the unpolarized parton distribution functions up to NNLO approximation from the QCD analysis of the world DIS data. To study the proton structure functions $F_2(x,Q^2)$ , we need to use the orthogonal polynomials expansion method. This method is very useful to parameterize parton distribution function at the input of $Q_0^2$ . Our calculations for parton distribution functions based on the Jacobi polynomials method are in good agreement with the other theoretical models.     

Solutions of several coupled discrete models in terms of Lamé polynomials of arbitrary order

Coupled discrete models are ubiquitous in a variety of physical contexts. We provide an extensive set of exact quasiperiodic solutions of a number of coupled discrete models in terms of Lamé polynomials of arbitrary order. The models discussed are: (i) coupled Salerno model, (ii) coupled Ablowitz?CLadik model, (iii) coupled ? ⁴ model and (iv) coupled ? ⁶ model. In all these cases we show that the coefficients of the Lamé polynomials are such that the Lamé polynomials can be re-expressed in terms of Chebyshev polynomials of the relevant Jacobi elliptic function.     

Classical Skew Orthogonal Polynomials and Random Matrices

Skew orthogonal polynomials arise in the calculation of the n-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the case that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed-form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre, and Jacobi cases.     

Variational principles for relativistic transport coefficients

Variational expressions for the transport coefficients of a one-component, relativistic gas are derived from the linearized relativistic Boltzmann equation for both quantum and classical gases. These expressions depend on functions _χ of the energy of the particles comprising the gas in such a way that: a) if _χ differs from a solution of the linearized Boltzmann equation by ε, then the value of the variational expression calculated with this _χ differs from the true value of the corresponding transport coefficient by ε²; and b) the value of the variational expression is always less than this true value. It is shown that values of the transport coefficients obtained by expanding _χ in a particular set of orthogonal polynomials and keeping only the first nontrivial term in the expansion are equivalent to those obtained using the Grad method of moments. It follows therefore that values obtained using this later method represent lower bounds on the true values. We also show that one can obtain simple, closed-form expressions for the various transport coefficients corresponding to an arbitrary number of terms in an expansion of the trial function _χ in the above-mentioned set of orthogonal polynomials. Finally we point out that all of our results can be carried over to the nonrelativistic case by taking the limit c → ∞.     

Nucleon/nuclei polarized structure function,using Jacobi polynomials expansion

We use the constituent quark model to extract polarized parton distributions and finally polarized nucleon structure function.Due to limited experimental data which do not cover whole(x,Q2)plane and to increase the reliability of the fitting,we employ the Jacobi orthogonal polynomials expansion.It will be possible to extract the polarized structure functions for Helium,using the convolution of the nucleon polarized structure functions with the light cone moment distribution.The results are in good agreement with available experimental data and some theoretical models.     

Bivariate distributions in statistical spectroscopy studies

Orthogonal polynomials in two variables, defined by a bivariate density function, are used to derive series expansions for expectation values with respect to the two variables. The convergence of the resulting polynomial expansion is due to the action of a central limit theorem. The shell model results for fixedE, J occupancies in (ds) ^m=5T=1/2 space are compared with the polynomial expansion results and the agreement is good.     

Beyond the first order of the Hyperspherical Harmonic Expansion Method

In this paper we demonstrate the inadequacy of the first order of the Hyperspherical Harmonic Expansion Method, the L_m approximation, for the calculation of the binding energies, charge form factors and charge densities of doubly magic nuclei like ¹⁶O and ⁴⁰Ca. We then extend the Hyperspherical Expansion Method to many-fermion systems, consisting of an arbitrary number of fermions, and develop an exact formalism capable of generating the complete optimal subset of the hyperspherical harmonic basis functions. This optimal subset consists of those hyperspherical harmonic basis functions directly connected to the dominant first term in the expansion, the hyperspherical harmonic of minimal order L_m, through the total interaction between the particles. The required many-body coefficients are given using either the Gogny or Talmi-Moshinsky coefficients for the two-body operators. Using the two-body coefficients the weight function generating the orthogonal polynomials associated with the optimal subset is constructed.     

The generic quantum superintegrable system on the sphere and Racah operators

We consider the generic quantum superintegrable system on the d-sphere with potential \(V(y)=\sum _{k=1}^{d+1}\frac{b_k}{y_k^2}\), where \(b_k\) are parameters. Appropriately normalized, the symmetry operators for the Hamiltonian define a representation of the Kohno–Drinfeld Lie algebra on the space of polynomials orthogonal with respect to the Dirichlet distribution. The Gaudin subalgebras generated by Jucys–Murphy elements are diagonalized by families of Jacobi polynomials in d variables on the simplex. We define a set of generators for the symmetry algebra, and we prove that their action on the Jacobi polynomials is represented by the multivariable Racah operators introduced in Geronimo and Iliev (Constr Approx 31(3):417–457, 2010). The constructions also yield a new Lie-theoretic interpretation of the bispectral property for Tratnik’s multivariable Racah polynomials.     

Baxter Operator Formalism for Macdonald Polynomials

We develop basic constructions of the Baxter operator formalism for the Macdonald polynomials associated with root systems of type A. Precisely, we construct a bispectral pair of mutually commuting Baxter operators such that the Macdonald polynomials are their common eigenfunctions. The bispectral pair of Baxter operators is closely related to the bispectral pair of recursive operators for Macdonald polynomials leading to various families of their integral representations. We also construct the Baxter operator formalism for the q-deformed ${\mathfrak{gl}_{\ell+1}}$ -Whittaker functions and the Jack polynomials obtained by degenerations of the Macdonald polynomials associated with the type A _? root system. This note provides a generalization of our previous results on the Baxter operator formalism for the Whittaker functions. It was demonstrated previously that Baxter operator formalism for the Whittaker functions has deep connections with representation theory. In particular, the Baxter operators should be considered as elements of appropriate spherical Hecke algebras and their eigenvalues are identified with local Archimedean L-factors associated with admissible representations of reductive groups over ${\mathbb{R}}$ . We expect that the Baxter operator formalism for the Macdonald polynomials has an interpretation in representation theory over higher-dimensional local/global fields.     

The scattering amplitude for a newly found exactly solvable potential

The scattering amplitude for a recently discovered exactly solvable shape invariant potential, which is isospectral to the generalized Pöschl–Teller potential, is calculated explicitly by considering the asymptotic behavior of the X₁$X_1$ Jacobi exceptional polynomials associated with this system.     

The Spectral Zeta Function for Laplace Operators on Warped Product Manifolds of the type I × f N

In this work we study the spectral zeta function associated with the Laplace operator acting on scalar functions defined on a warped product of manifolds of the type I × _f N, where I is an interval of the real line and N is a compact, d-dimensional Riemannian manifold either with or without boundary. Starting from an integral representation of the spectral zeta function, we find its analytic continuation by exploiting the WKB asymptotic expansion of the eigenfunctions of the Laplace operator on M for which a detailed analysis is presented. We apply the obtained results to the explicit computation of the zeta regularized functional determinant and the coefficients of the heat kernel asymptotic expansion.