Part Ⅰ Greens Functions in Mathematical Physics
1 Time-Independent Greens Functions
1.1 Formalism
1.2 Examples
1.2.1 Three-Dimensional Case (d=3)
1.2.2 Two-Dimensional Case (d=2)
1.2.3 One-Dimensional Case (d=1)
1.2.4 Finite Domain (2
1.3 Summary
1.3.1 Definition
1.3.2 Basic Properties
1.3.3 Methods of Calculation
1.3.4 Use
Further Reading
Problems
2 Time-Dependent Greens Functions
2.1 First-Order Case
2.1.1 Examples
2.2 Second-Order Case
2.2.1 Examples
2.3 Summary
2.3.1 Definition
2.3.2 Basic Properties
2.3.3 Definition
2.3.4 Basic Properties
2.3.5 Use
Further Reading
Problems

Part Ⅱ Greens Functions in One-Body Quantum Problems
3 Physical Significance of G.Application to the Free-Particle Case
3.1 General Relations
3.2 The Free-Particle (Ho=p2/2m) Case
3.2.1 3-d Case
3.2.2 2-d Case
3.2.3 1-d Case
3.3 The Free-Particle Klein Gordon Case
3.4 Summary
Further Reading
Problems
4 Greens Functions and Perturbation Theory
4.1 Formalism
4.1.1 Time-Independent Case
4.1.2 Time-Dependent Case
4.2 Applications
4.2.1 Scattering Theory (E>0)
4.2.2 Bound State in Shallow Potential Wells (E<0)
4.2.3 The KKR Method for Electronic Calculations in Solids.
4.3 Summary
Further Reading
Problems
5 Greens Functions for Tight-Binding Hamiltonians
5.1 Introductory Remarks
5.2 The Tight-Binding Hamiltonian (TBH)
5.3 Greens Functions
5.3.1 One-Dimensional Lattice
5.3.2 Square Lattice
5.3.3 Simple Cubic Lattice
5.3.4 Greens Functions for Bethe Lattices (Cayley Trees)
5.4 Summary
Further Reading
Problems
6 Single Impurity Scattering
6.1 Formalism
6.2 Explicit Results for a Single Band
6.2.1 Three-Dimensional Case
6.2.2 Two-Dimensional Case
6.2.3 One-Dimensional Case
6.3 Applications
6.3.1 Levels in the Gap
6.3.2 The Cooper Pair and Superconductivity
6.3.3 The Kondo Problem
6.3.4 Lattice Vibrations in Crystals Containing "Isotope" Impurities
6.4 Summary
Further Reading
Problems
7 Two or More Impurities; Disordered Systems
7.1 Two Impurities
7.2 Infinite Number of Impurities
7.2.1 Virtual Crystal Approximation (VCA)
7.2.2 Average t-Matrix Approximation (ATA)
7.2.3 Coherent Potential Approximation (CPA)
7.2.4 The CPA for Classical Waves
7.2.5 Direct Extensions of the CPA
7.2.6 Cluster Generalizations of the CPA
7.3 Summary
Further Reading
Problems
8 Electrical Conductivity and Greens Functions
8.1 Electrical Conductivity and Related Quantities
8.2 Various Methods of Calculation
8.2.1 Phenomenological Approach
8.2.2 Boltzmanns Equation
8.2.3 A General， Independent-Particle Formula for Conductivity
8.2.4 General Linear Response Theory
8.3 Conductivity in Terms of Greens Functions
8.3.1 Conductivity Without Vertex Corrections
8.3.2 CPA for Vertex Corrections
8.3.3 Vertex Corrections Beyond the CPA
8.3.4 Post-CPA Corrections to Conductivity
8.4 Summary
Further Reading
Problems
9 Localization， Transport， and Greens Functions
9.1 An Overview
9.2 Disorder， Diffusion， and Interference
9.3 Localization
9.3.1 Three-Dimensional Systems
9.3.2 Two-Dimensional Systems
9.3.3 One-Dimensional and Quasi-One-Dimensional Systems
9.4 Conductance and Transmission
9.5 Scaling Approach
9.6 Other Calculational Techniques
9.6.1 Quasi-One-Dimensional Systems and Scaling
9.6.2 Level Spacing Statistics
9.7 Localization and Greens Functions
9.7.1 Greens Function and Localization in One Dimension .
9.7.2 Renormalized Perturbation Expansion (RPE) and Localization
9.7.3 Greens Functions and Transmissions in Quasi-One-Dimensional Systems
9.8 Applications
9.9 Summary
Further Reading
Problems

Part Ⅲ Greens Functions in Many-Body Systems
10 Definitions
10.1 Single-Particle Greens Functions in Terms of Field Operators
10.2 Greens Functions for Interacting Particles
10.3 Greens Functions for Noninteracting Particles
10.4 Summary
Further Reading
Problems
11 Properties and Use of the Greens Functions
11.1 Analytical Properties of gs and gs
11.2 Physical Significance and Use of gs and gs
11.3 Quasiparticles
11.4 Summary
11.4.1 Properties
11.4.2 Use
Further Reading
Problems
12 Calculational Methods for g
12.1 Equation of Motion Method
12.2 Diagrammatic Method for Fermions at T=0
12.3 Diagrammatic Method for T≠0
12.4 Partial Summations. Dysons Equation
12.5 Other Methods of Calculation
12.6 Summary
Further Reading
Problems
13 Applications
13.1 Normal Fermi Systems. Landau Theory
13.2 High-Density Electron Gas
13.3 Dilute Fermi Gas
13.4 Superconductivity
13.4.1 Diagrammatic Approach
13.4.2 Equation of Motion Approach
13.5 The Hubbard Model
13.6 Summary
Further Reading
Problems
A Diracs delta Function
B Diracs bra and ket Notation
C Solutions of Laplace and Helmholtz Equations in Various Coordinate Systems
C.1 Helmholtz Equation
C.1.1 Cartesian Coordinates
C.1.2 Cylindrical Coordinates
C.1.3 Spherical coordinates
C.2 Vector Derivatives
C.2.1 Spherical Coordinates
C.2.2 Cylindrical Coordinates
C.3 Schrodinger Equation in Centrally Symmetric 3-and 2-Dimensional Potential V
D Analytic Behavior of G(z) Near a Band Edge
E Wannier Functions
F Renormalized Perturbation Expansion (RPE)
G Boltzmanns Equation
H Transfer Matrix， S-Matrix， etc
I Second Quantization
Solutions of Selected Problems
References
Index