scholarly journals Some Connections between the Spherical and Parabolic Bases on the Cone Expressed in terms of the Macdonald Function

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
I. A. Shilin ◽  
Junesang Choi
Keyword(s):  
Linear Operator ◽  
Hypergeometric Functions ◽  
Legendre Function ◽  
Integral Formula ◽  
Macdonald Function ◽  
Representation Space ◽  
Legendre Functions ◽  
Matrix Elements ◽  
The Matrix ◽  
Spherical Basis

Computing the matrix elements of the linear operator, which transforms the spherical basis ofSO(3,1)-representation space into the hyperbolic basis, very recently, Shilin and Choi (2013) presented an integral formula involving the product of two Legendre functions of the first kind expressed in terms of 4F3-hypergeometric function and, using the general Mehler-Fock transform, another integral formula for the Legendre function of the first kind. In the sequel, we investigate the pairwise connections between the spherical, hyperbolic, and parabolic bases. Using the above connections, we give an interesting series involving the Gauss hypergeometric functions expressed in terms of the Macdonald function.

scholarly journals COHERENT STATES, DISPLACED NUMBER STATES AND LAGUERRE POLYNOMIAL STATES FOR su(1, 1) LIE ALGEBRA

2000 ◽  
Vol 14 (10) ◽  
pp. 1093-1103 ◽  
Author(s):  
XIAO-GUANG WANG
Keyword(s):  
Coherent State ◽  
Lie Algebra ◽  
Hypergeometric Functions ◽  
Laguerre Polynomial ◽  
Optical Systems ◽  
Matrix Elements ◽  
Operator Formalism ◽  
Ladder Operator ◽  
Deformed Oscillator ◽  
The Matrix

The ladder operator formalism of a general quantum state for su(1, 1) Lie algebra is obtained. The state bears the generally deformed oscillator algebraic structure. It is found that the Perelomov's coherent state is a su(1, 1) nonlinear coherent state. The expansion and the exponential form of the nonlinear coherent state are given. We obtain the matrix elements of the su(1, 1) displacement operator in terms of the hypergeometric functions and the expansions of the displaced number states and Laguerre polynomial states are followed. Finally some interesting su(1, 1) optical systems are discussed.

HYPERGEOMETRIC FUNCTION REPRESENTATION OF RELATIVISTIC TRANSITION MATRIX ELEMENTS FOR HYDROGENIC ATOMS

2009 ◽  
Vol 23 (02) ◽  
pp. 111-119
Author(s):  
A. V. SOLDATOV ◽  
J. SEKE ◽  
G. ADAM ◽  
M. POLAK
Keyword(s):  
Transition Matrix ◽  
Numerical Evaluation ◽  
Hypergeometric Functions ◽  
Field Vector ◽  
Matrix Elements ◽  
Plane Wave Expansion ◽  
Computationally Efficient ◽  
The Matrix ◽  
Transition Matrix Elements ◽  
Computationally Efficient Algorithms

By using the plane-wave expansion for the electromagnetic-field vector potential, relativistic bound–bound, bound–unbound and unbound–unbound transition matrix elements for hydrogenic atoms are expressed universally in terms of hypergeometric functions. By applying the obtained formulas, these transition matrix elements can be evaluated analytically and numerically with arbitrarily high precision. The newfound representation for the matrix elements is very convenient for direct numerical evaluation of the Lamb shift because of its universality, conciseness and reliance on functions already built in the standard computational packages. All of this is highly favorable for programming of computationally efficient algorithms.

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

scholarly journals Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

2021 ◽  
Author(s):  
Mariusz Pawlak ◽  
Marcin Stachowiak
Keyword(s):  
Spherical Harmonics ◽  
Interaction Potential ◽  
Chem Phys ◽  
Basis Set ◽  
Matrix Elements ◽  
Trigonometric Functions ◽  
Variational Theory ◽  
Molecular Collision ◽  
The Matrix ◽  
Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

On the matrix versions of Appell hypergeometric functions

2014 ◽  
Vol 37 (1) ◽  
pp. 31-38 ◽  
Author(s):  
Abdullah Altin ◽  
Bayram Çekіm ◽  
Recep Şahіn
Keyword(s):  
Hypergeometric Functions ◽  
The Matrix

Notizen: The Dipole Moment Function of HF Molecule Using Morse Potential

1977 ◽  
Vol 32 (8) ◽  
pp. 897-898 ◽  
Author(s):  
Y. K. Chan ◽  
B. S. Rao
Keyword(s):  
Wave Equation ◽  
Dipole Moment ◽  
Potential Function ◽  
Electric Dipole ◽  
Morse Potential ◽  
Moment Function ◽  
Matrix Elements ◽  
The Matrix ◽  
Vibration Rotation ◽  
Experimental Values

Abstract The radial Schrödinger wave equation with Morse potential function is solved for HF molecule. The resulting vibration-rotation eigenfunctions are then used to compute the matrix elements of (r - re)n. These are combined with the experimental values of the electric dipole matrix elements to calculate the dipole moment coefficients, M 1 and M 2.

Leach Behavior of SRL TDS-131 Defense Waste Glass in Water at High&Low Flow Rates

1983 ◽  
Vol 26 ◽  
Author(s):  
Aaron Barkatt ◽  
William Sousanpour ◽  
Alisa Barkatt ◽  
Morad A. Boroomand ◽  
Pedro B. Macedo
Keyword(s):  
Contact Time ◽  
Protective Layer ◽  
High Flow ◽  
Low Flow ◽  
Matrix Elements ◽  
Flow Rates ◽  
Low Concentrations ◽  
Controlling Mechanism ◽  
The Matrix ◽  
Leach Behavior

ABSTRACTLeach tests carried out on SRL TDS-131 Defense Waste Class indicate that at high flow rates the controlling mechanism is simple corrosion. The matrix elements (Si, Al) are leached out at rates similar to those of the leaching of the alkalis and of boron, and the leaching process is nearly linear with time. At slow flow rates (below 1 m/yr) leaching becomes controlled by the build-up of a protective layer. Al and most of the Si remain in the leached surface layer. The leach rates decrease in the course of the test before leveling off at constant values which are almost inversely proportional to the contact time, indicating that leachate concentrations have become solubility-limited. The low concentrations observed at this stage indicate the formation of alteration products.

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