Jost Functions in Quantum Mechanics: A Unified Approach to Scattering, Bound, and Resonant State Problems

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