Multivariate Hypergeometric Functions Related to the Coulomb Problem

This work deals with several aspects of the Coulomb problem as well as with a large number of related multivariable hypergeometric functions. Intended for atomic physicists and mathematicians, it builds a bridge of understanding between both the Physical and Mathematical sciences. After a basic introduction on the Coulomb problem, it is divided into two parts. First, an original study of the two- and three-body Coulomb problem is presented with emphasis on related mathematical issues. Most of the aspects of this part have the common point that they involve a significant number of multivariable hypergeometric functions. These are studied in the second, mathematical, part of the book where emphasis is put on the relevant expressions for physical applications. This work will be an important tool in for atomic physicists dealing with two- and three-body Coulomb problems and includes aspects not addressed elsewhere. A significant number of hypergeometric functions in two, three and morevariables related to physical problems are presented and studied. The aspect of considering both mathematical and physical aspects in connection with hypergeometric functions makes the book unique.

Part I

Atomic Physics

1 Basic elements of atomic physics

1.1 The Schrödinger equation for two- and three-body problems.

Coordinate systems. Angular momentum representations.

1.2 Bound and continuum states: asymptotic behaviors.

1.3 Bound states.

1.3.1 Variational methods.

1.3.2 Basis set for bound states.

1.4 Continuum states.

1.4.1 The non-homogeneous Schrödinger equation.

1.4.2 The Lippmann-Schwinger equation.

1.4.3 Perturbative series.

1.4.4 Basis set expansions.

2 The two-body Coulomb problem

2.1 Closed form solutions for the Coulomb potential in spherical

coordinates.

2.1.1 Homogeneous radial Coulomb wave equation and its solutions. Regular and

irregular solutions. Asymptotic behaviors.

2.1.2 Non-homogeneous radial Coulomb wave equations. Solutions regular at the

origin and with different asymptotic behaviors. L2 representations and its

connection with the J-matrix methods for scattering process studies. The

off-shell solution for the Coulomb problem.

2.1.3 Screened Coulomb potentials. Transformation of the Schrödinger equation

into a system of non-homogeneous ordinary differential equations. Solutions

to the equations.

2.2 Closed form solutions for the Coulomb potential in parabolic

coordinates.

2.2.1 The homogeneous Coulomb wave equation and its solutions. Regular and

irregular solutions. Asymptotic behaviors.

2.2.2 The Born series for the wavefunction. The Born series for the transition

amplitude.

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Table of contents

2.3 Sturmian functions for the Coulomb problem.

2.3.1 Applications to the solution of bound states problems. Applications to

scattering problems. Representation for the Coulomb Greens function.

3 The three-body Coulomb problem

3.1 The Schrödinger equation in interparticle coordinates.

3.1.1 Dependence on the masses of the particles. Spherical coordinates. Gener-

alized parabolic coordinates.

3.2 The product of three two-body Coulomb wave functions: C3

approach.

3.2.1 Solutions in spherical coordinates. Solutions in generalized parabolic coor-

dinates. Asymptotic behaviors.

3.3 Generalizations of the C3 model based on effective parameters.

3.3.1 Velocity dependent charges. Coordinate dependent charges. Asymptotic

models.

3.4 Generalizations of the C3 model based on multivariable hyper-

geometric functions.

3.4.1 Models based on the Phi 2 hypergeometric function. The Burchnall and Chaundy

operator as a Green's operator. The Phi 0 model.

3.5 Configuration Interaction with angular correlation: spherical

coordinates representation.

3.5.1 Eigenenergy basis for bound and continuum states. Implementation of the

Configuration Interaction method. Representation of the hamiltonian and

overlapping matrix elements in terms of multivariable hypergeometric func-

tions. Applications to the calculation of bound states of the Helium atom

and other Helium like ions.

3.5.2 Coulomb Sturmian basis functions for bound and continuum states. Con-

¯guration Interaction method. Scattering states. L2 representations.

3.6 Configuration Interaction with angular correlation: parabolic

coordinates representation.

3.6.1 Eigenenergy basis for bound and continuum states. Implementation of the

Configuration Interaction method for scattering states.

3.6.2 Coulomb Sturmian basis functions. Configuration Interaction method. Scat-

tering states. L2 representations.

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Contents

Part II

Hypergeometric Functions

4 One-variable hypergeometric functions

4.1 Introduction.

4.2 The Kummer function.

4.2.1 Power series expansion.

4.2.2 Differential equation.

4.2.3 Integral representations.

4.2.4 Asymptotic limits.

4.3 The Gauss function.

4.3.1 Power series expansion.

4.3.2 Differential equation.

4.3.3 Integral representations.

4.4 Non-homogeneous Kummer differential equations.

4.4.1 Power series expansion.

4.4.2 Integral representations.

4.5 Non-homogeneous Gauss differential equations.

4.5.1 Power series expansion.

4.5.2 Integral representations.

4.6 Generalized hypergeometric functions.

5 Two-variables hypergeometric functions

5.1 General considerations.

5.1.1 General considerations. The Appell technique. The Appell functions. Gen-

eral properties. System of partial differential equations. Integral represen-

tations. Series representations in terms of other hypergeometric functions.

5.2 Confluent functions.

5.2.1 Functions associated to non-homogeneous two-body Coulomb problems.

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Contents

5.2.2 The Phi 2 Erdélyi function. System of partial differential equations. Integral

representations. Representations in terms of simpler functions.

5.2.3 The 1 theta (1)

1 function. General properties. System of partial differential equa-

tions. Integral representations. Representations in terms of simpler func-

tions. The ordinary differential equation associated.

5.3 Non confluent functions.

5.3.1 The 2 theta (1)

1 function. General properties. System of partial differential equa-

tions. Integral representations. Representations in terms of simpler func-

tions. The associated ordinary differential equation.

5.3.2 The 2 Lambda (2) functions. System of partial differential equations. Integral repre-

sentations.

6 Three-variables hypergeometric functions

6.1 Confluent functions.

6.1.1 The Phi 3 function. System of differential equations. Integral representations.

Representations in terms of simpler functions.

6.1.2 The Phi 0 function. System of differential equations. Integral representations.

Representations in terms of simpler functions.

6.1.3 The 1 Theta (2)

1 function. General properties. System of partial differential equa-

tions. Integral representations. Representations in terms of simpler func-

tions. The associated ordinary differential equation.

6.1.4 The ¥ function. General properties. System of partial differential equations.

Integral representations. Representations in terms of simpler functions.

The associated ordinary differential equation.

6.2 Non confluent functions.

6.2.1 The Lamdba (3) functions. General properties. System of partial differential equa-

tions. Integral representations. Series representations in terms of other

hypergeometric functions.

6.2.2 The ¢(3) function. General properties. System of partial di®erential equa-

tions. Integral representations. Series representations in terms of other

hypergeometric functions.

7 n-variables hypergeometric functions

7.1 Congruent functions.

7.1.1 The 1Theta (n)

1 function. General properties.

7.2 Non confluent functions.

7.2.1 The 2Theta (n) 1 function. General properties.

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Contents

7.2.2 The Lamdba (n) functions. General properties.

7.2.3 The Delta (n) function. General properties.

8 Miscellaneous

8.1 Burchnall and Chaundy operators.

8.1.1 The Delat  and Delta-1 operators. Application on confluent hypergeometric functions.

Generalizations. Other hypergeometric functions obtained from generalized

operators.

Bibliography

Index