Based on Jost function theory this book presents an approach useful for different types of quantum mechanical problems. These include the description of scattering, bound, and resonant states, in a unified way. The reader finds here all that is known about Jost functions as well as what is needed to fill the gap between the pure mathematical theory and numerical calculations. Some of the topics covered are: quantum resonances, Regge poles, multichannel scattering, Coulomb interaction, Riemann surfaces, multichannel analog of the effective range theory, one- and two-dimensional problems, many-body problems within the hyperspherical approach, just to mention few of them. These topics are relevant in the fields of quantum few-body theory, nuclear reactions, atomic collisions, and low-dimensional semiconductor nanostructures. In light of this, the book is meant for students, who study quantum mechanics, scattering theory, or nuclear reactions at the advanced level as well as for post-graduate students and researchers in the fields of nuclear and atomic physics. Many of the arguments that are traditional for textbooks on quantum mechanics and scattering theory, are covered here in a different way, using the Jost functions. This gives the reader a new insight into the subject, revealing new features of various mathematical objects and quantum phenomena.
Author(s): Sergei A. Rakityansky
Publisher: Springer
Year: 2022
Language: English
Pages: 634
City: Cham
Preface
Contents
1 The Basic Concepts
1.1 Quantum Vectors
1.2 Schrödinger Equation
1.3 Boundary Conditions
1.3.1 Bound States
1.3.2 Scattering States
1.3.2.1 Plane and Spherical Waves
1.3.2.2 Scattering Amplitude and Cross Section
1.3.3 Resonances
1.4 Semi-classical Wave Function
1.5 Two-Body Problem
Part I Single-Channel Problems
2 Schrödinger Equation and Its Solutions
2.1 Regular and Irregular Solutions of the Radial Equation
2.2 Finite-Range Potential
2.2.1 Transformation of Schrödinger Equation
2.2.1.1 First-Order Differential Equations
2.2.1.2 Alternative Form of the Differential Equations
2.2.1.3 Integral Equations
2.2.2 Uniform Bound for the Regular Solution
2.2.2.1 Integral Equation for the Regular Solution
2.2.2.2 Upper Bound for the Regular Solution
2.2.3 Jost Functions
2.2.4 Jost Solutions
2.2.4.1 Integral Equations
2.2.5 Analyticity of the Jost Functions
3 Riemann Surface and the Spectral Points
3.1 Symmetry Properties of the Jost Functions
3.1.1 Vertical Symmetry
3.1.2 Diagonal Symmetry
3.1.3 Horizontal Symmetry
3.2 High-Energy Asymptotics of the Jost Functions
3.3 Spectral Points
3.3.1 Bound States
3.3.2 Resonances
3.3.3 Virtual States and Sub-threshold Resonances
3.3.4 Resonance Wave Function Normalization
3.3.5 Simplicity of the Bound and Resonant State Zeros
3.3.6 Spectral Point at Threshold Energy
3.3.6.1 Multiplicity of Threshold Zeros
3.3.7 Distribution of Spectral Points Over theRiemann Surface
3.3.8 Number of Spectral Points
3.3.9 Integral Equation for a Bound-State Wave Function
3.3.10 Bargmann's Inequality
4 Scattering States and the S-Matrix
4.1 Partial Waves
4.2 S-Matrix
4.3 Phase Shift
4.4 Resonant Scattering
4.5 Breit-Wigner Resonances
4.6 Analytic Properties of the S-Matrix
4.6.1 Symmetry of the S-Matrix
4.6.2 Spectral Expansion of the S-Matrix
4.6.3 Residues of the S-Matrix and the ANC
4.6.4 Argand Plot
4.6.5 Causality and Analyticity
4.7 Levinson's Theorem
5 Complex Angular Momentum
5.1 Symmetry Properties of the Jost Functions
5.2 Regge Poles
5.3 Simplicity of Regge Zeros
5.3.1 Asymptotic Normalization Constant (ANC)
5.4 Regge Trajectories
5.5 Regge Poles and Resonance Parameters
5.6 Watson Transform
6 Green's Functions
6.1 Free Green's Function for Scattering Solution
6.2 Total Green's Function for Scattering Solution
6.3 Free and Total Green's Function for the Regular Solution
6.4 Free and Total Green's Function for the Jost Solutions
6.5 Three-Dimensional Free Green's Function
6.6 Summary
6.7 Jost–Pais Theorem
7 Short-Range Potential Extending to Infinity
7.1 The Regular Solution
7.1.1 Long-Range Asymptotics
7.1.1.1 WKB Asymptotic Analysis
7.2 Jost Functions
7.2.1 Incoming and Outgoing Waves at Complex Momenta
7.2.2 Exponentially Decaying Potentials
7.3 Analyticity of the Jost Functions
7.4 Complex Rotation
7.4.1 Exponentially Decaying Potentials
7.4.2 Non-analytic Potentials
7.5 Redundant Poles of the S-Matrix
7.6 From Finite-Range to Short-Range Potentials
7.7 Analytic Structure of the Jost Functions
7.7.1 Factorization
7.7.2 Domain of Analyticity and Complex Rotation
7.8 Generalized Levinson's theorem
7.9 Dispersion Relations
8 Single-Channel Potential with Coulombic Tail
8.1 Pure Coulomb Potential
8.1.1 Schrödinger Equation
8.1.2 Jost Functions
8.1.3 Scattering
8.1.3.1 Gamow Factor
8.2 Short-Range Plus Coulomb Potential
8.2.1 First-Order Differential Equations
8.2.2 Integral Equations
8.2.3 Jost Functions
8.2.4 Jost Solutions
8.2.5 Analyticity of the Jost Functions
8.2.6 Analytic Structure of the Jost Functions
8.2.7 Domain of Analyticity and Complex Rotation
8.2.8 Short-Range Plus Attractive Coulomb Potential
8.2.9 Riemann Surface for a System of Charged Particles
8.2.10 Symmetry Properties of the Jost Functions
8.2.11 Asymptotic Normalization Constant (ANC)
Part II Multi-Channel Problems
9 Non-central Potential
9.1 Partial Waves
9.1.1 Discrete States
9.1.2 Scattering States
9.2 Fundamental Matrix of Regular Solutions
9.3 Transformation of Schrödinger Equation
9.3.1 Incoming and Outgoing Waves
9.3.2 First-Order Differential Equations
9.3.3 Alternative Form of the Differential Equations
9.3.4 Boundary Conditions
9.3.4.1 Explicit Behaviour of the Regular Solution Near r=0
9.3.4.2 Cancellation of Singularities Near r=0
9.3.4.3 Combination of the Second- and the First-Order Equations
9.4 Asymptotic Behaviour of the Fundamental Matrix
9.5 Jost Matrices
9.5.1 Complex Rotation
9.6 Jost Solutions
9.6.1 Jost Solutions Near r=0
9.7 Physical Solutions
9.7.1 Bound States and Resonances
9.7.2 Scattering States
10 Systems with Non-zero Spins
10.1 Spin-Angular State-Vectors
10.2 Partial-Wave Decomposition for Discrete Spectrum
10.2.1 Radial Schrödinger Equation
10.3 Partial-Wave Decomposition for Scattering States
10.3.1 Radial Schrödinger Equation
10.3.2 Plane Wave with Non-zero Spin
10.3.3 Long-Range Asymptotics of the Wave Function
10.3.4 Scattering Observables
10.4 Symmetries of the Jost Matrices
10.5 Analytic Structure of the Jost Matrices
10.6 Time-Reversal Invariance, Unitarity, and Parity Conservation
10.6.1 Time-Reversal Invariance
10.6.2 Unitarity
10.6.3 Conservation of Parity
10.6.4 Reciprocity and Detailed Balance
10.7 Simplicity of the Bound and Resonant State Zeros
10.8 Asymptotic Normalization Constants
10.9 Scattering Phase Shifts
10.9.1 Example: Two Coupled Partial Waves
10.9.1.1 Eigen-Phase Shifts
10.9.1.2 Nuclear-Bar Phase Shifts
11 Multi-Channel Schrödinger Equation
11.1 Channels with Different Types of Particles
11.2 Partial-Wave Decomposition
11.2.1 Spin-Angular Matrices
11.2.2 Partial-Wave Decomposition for Discrete Spectrum
11.2.2.1 Radial Schrödinger Equation
11.2.3 Partial-Wave Decomposition for Scattering States
11.2.3.1 Radial Schrödinger Equation
12 Multi-Channel Jost Matrix
12.1 First-Order Differential Equations
12.2 Complex Rotation
12.3 Jost Solutions
12.4 Spectral Points
12.4.1 Bound States
12.4.2 Resonances and Their Partial Decay Widths
12.4.3 Simplicity of the Spectral Point Zeros and the ANC
12.5 Multi-Channel Scattering
12.6 Scattering Observables
12.7 Unitarity, Reciprocity, and Detailed Balance
13 Riemann Surfaces for Multi-Channel Systems
13.1 Cuts and Interconnections
13.2 Degenerate Channels
13.3 Distribution of the Spectral Points
13.4 Analytic Structure of the Jost Matrices
13.5 Symmetry Properties of the Jost Matrices
14 Multi-channel Problems of Charged Particles
14.1 Jost Matrix
14.2 Jost Solutions
14.3 Complex Rotation
14.4 Simplicity of the Spectral Point Zeros and the ANC
14.5 Scattering Observables
14.6 Analytic Structure of the Jost Matrices
14.7 Attractive Coulomb Forces
14.8 Riemann Surfaces
14.9 Symmetry Properties of the Jost Matrices
15 Effective-Range Expansion and Its Generalizations
15.1 Single-Channel Short-Range Potential
15.1.1 Effective-Range Expansion
15.1.2 Expansion Coefficients: Calculation
15.1.3 Constructing Potentials with Given Properties
15.1.4 Expansion Coefficients: Fitting the Data
15.2 Single-Channel Coulomb-Tailed Potential
15.2.1 Expansion Coefficients
15.3 Multi-channel short-Range Potential
15.4 Multi-channel Coulomb-Tailed Potential
Part III Special Issues
16 Singular and Low-Dimensional Potentials
16.1 Singular Potential
16.1.1 Boundary Conditions
16.2 One-Dimensional Problems
16.2.1 Schrödinger Equation for a 1D System
16.2.2 Boundary Conditions
16.2.3 Jost Matrices
16.2.4 Riemann Surface
16.2.5 Spectral Points
16.2.6 Scattering
16.3 Two-Dimensional Problems
16.3.1 Partial-Wave Decomposition
16.3.2 Jost Functions
16.3.3 Analytic Properties of the Jost Functions
16.3.4 Power-Series Expansions
16.3.5 2D-Scattering
16.3.5.1 Plane and Circular Waves
16.3.5.2 Scattering Wave Function
16.3.5.3 Cross Section
17 Miscellaneous Extensions of the Jost Function Approach
17.1 Many-Body Problems
17.1.1 Hyperspherical Expansion
17.1.2 Hyperradial Equation and the Jost Matrices
17.1.3 Analytic Structure of the Jost Matrices
17.2 Non-local Potential
17.2.1 Schrödinger Equation
17.2.2 Non-locality and Velocity Dependence
17.2.3 Jost Functions
17.2.3.1 Short-Distance Behaviour of Non-local Potentials
17.2.3.2 Symmetry and Analytic Structure of the Jost Functions
17.2.4 Separable Potential
17.2.5 Generalized Jost–Pais Theorem
17.3 Off-Shell Jost Functions
18 Some Exactly Solvable Potential Models
18.1 Exponential Potential
18.2 Single-Channel Square-Well Potential
18.2.1 Analytic Structure
18.3 Single-Channel Square-Well Potential with the Coulomb Tail
18.3.1 Analytic Structure
18.4 Two-Channel Square-Well Potential
18.4.1 Analytic Structure
18.5 Two-Channel Square-Well Potential with the Coulomb Tail
18.5.1 Analytic Structure
18.6 Bargmann Potentials
18.6.1 ``Linear'' Type Bargmann Potentials
18.6.1.1 Potentials with r-2 Tail
18.6.1.2 Eckart Potential
A Partial-Wave Expansion
B Basics of Complex Analysis
C Wronskian of Scalar and Matrix Functions
D Bessel Functions
D.1 Definition and Some Properties
D.2 Analytic Structure of j(kr) and n(kr)
D.2.1 Integer Order
D.2.2 Half-Integer Order
E Coulomb Wave Functions
E.1 Definitions and the Main Properties
E.2 Analytic Structure
E.3 Coulomb-Related Functions Near the Spectral Points
F Integral Equations
F.1 Separable Kernel
F.2 Numerical Solution
F.3 Fredholm Theory
F.3.1 Fredholm Alternative
F.3.2 Fredholm Determinant and Resolvent
F.4 Contraction Mapping Principle
F.5 Contraction Mapping for Fredholm Equation
F.6 Contraction Mapping for Volterra Equation
G Poincaré Theorem
H Newton Method for Locating Zeros of a Complex Function
I Choice of the Units
I.1 Nuclear Units
I.2 Atomic Units
Bibliography
Bibliography
Index
