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Register a new userAlgebraic evaluation of matrix elements in the Laguerre function basis
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The Laguerre functions constitute one of the fundamental basis sets for calculations in atomic and molecular electron-structure theory, with applications in hadronic and nuclear theory as well. While similar in form to the Coulomb bound-state eigenfunctions (from the Schroedinger eigenproblem) or the Coulomb-Sturmian functions (from a related Sturm-Liouville problem), the Laguerre functions, unlike these former functions, constitute a complete, discrete, orthonormal set for square-integrable functions in three dimensions. We construct the SU(1,1)xSO(3) dynamical algebra for the Laguerre functions and apply the ideas of factorization (or supersymmetric quantum mechanics) to derive shift operators for these functions. We use the resulting algebraic framework to derive analytic expressions for matrix elements of several basic radial operators (involving powers of the radial coordinate and radial derivative) in the Laguerre function basis. We illustrate how matrix elements for more general spherical tensor operators in three dimensional space, such as the gradient, may then be constructed from these radial matrix elements.
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Multipole matrix elements of Green function of Laplace equation are calculated. The multipole matrix elements of Green function in electrostatics describe potential on a sphere which is produced by a charge distributed on the surface of a different (possibly overlapping) sphere of the same radius. The matrix elements are defined by double convolution of two spherical harmonics with the Green function of Laplace equation. The method we use relies on the fact that in the Fourier space the double convolution has simple form. Therefore we calculate the multipole matrix from its Fourier transform. An important part of our considerations is simplification of the three dimensional Fourier transformation of general multipole matrix by its rotational symmetry to the one-dimensional Hankel transformation.
The new complete orthonormal sets of -Laguerre type polynomials (-LTP,) are suggested. Using Schrodinger equation for complete orthonormal sets of -exponential type orbitals (-ETO) introduced by the author, it is shown that the origin of these polynomials is the centrally symmetric potential which contains the core attraction potential and the quantum frictional potential of the field produced by the particle itself. The quantum frictional forces are the analog of radiation damping or frictional forces suggested by Lorentz in classical electrodynamics. The new -LTP are complete without the inclusion of the continuum states of hydrogen like atoms. It is shown that the nonstandard and standard conventions of -LTP and their weight functions are the same. As an application, the sets of infinite expansion formulas in terms of -LTP and L-Generalized Laguerre polynomials (L-GLP) for atomic nuclear attraction integrals of Slater type orbitals (STO) and Coulomb-Yukawa like correlated interaction potentials (CIP) with integer and noninteger indices are obtained. The arrange and rearranged power series of a general power function are also investigated. The convergence of these series is tested by calculating concrete cases for arbitrary values of parameters of orbitals and power function.
Symbolic methods of umbral nature play an important and increasing role in the theory of special functions and in related fields like combinatorics. We discuss an application of these methods to the theory of lacunary generating functions for the Laguerre polynomials for which we give a number of new closed form expressions. We present furthermore the different possibilities offered by the method we have developed, with particular emphasis on their link to a new family of special functions and with previous formulations, associated with the theory of quasi monomials.
In this paper, we present an eleven invariant isotropic irreducible function basis of a third order three-dimensional symmetric tensor. This irreducible function basis is a proper subset of the Olive-Auffray minimal isotropic integrity basis of that tensor. The octic invariant and a sextic invariant in the Olive-Auffray integrity basis are dropped out. This result is of significance to the further research of irreducible function bases of higher order tensors.
We suggest the modified matching procedure for TMD PDF to the integrated PDF aimed to increase the amount of perturbative information in the TMD PDF expression. The procedure consists in the selection and usage of the non-minimal operator basis, which restricts the expansion to desired general behavior. The implication of OPE allows to systematic account of the higher order corrections. In the case of TMD PDF we assume the Gaussian behavior, which suggests Laguerre polynomial basis as the best for the convergence of OPE. We present the leading and next-to-leading expression of TMD PDF in this basis. The obtained perturbative expression for the TMD PDF is valid in the wide region of $b_T$ (we estimate this region as $b_Tlesssim 2-3$ GeV$^{-1}$ depending on $x$).
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