CONTENTS

1  INTRODUCTION 

1-1 Calculational Techniques in Condensed Matter Theory
1-2 Recursion Method Applied to Many-Body Dynamics
1-3 Formalism and Goals
1-4 Applications 
 

2 LINEAR RESPONSE AND EQUILIBRIUM DYNAMICS

2-1 Response Function and Generalized Susceptibility
2-2 Fluctuation-Dissipation Theorem
2-3 Moment Expansion 
 

3 LIOUVILLIAN REPRESENTATION 

3-1 Quantum Formulation
3-2 Classical Formulation
3-3 Orthogonal Expansion of Dynamical Variables
3-4 Relaxation Function and Spectral Density
3-5 Recursion Method and Moment Expansion
3-6 Generalized Langevin Equation
3-7 Projection Operator Formalism
3-8 Retarded Green's Functions
 

4 HAMILTONIAN REPRESENTATION 

4-1 Orthogonal Expansion of Wave Functions
4-2 Structure Function
4-3 Continued-Fraction Coefficients and Frequency Moments
4-4 Lanczos Algorithm
4-5 Modified Lanczos Method
4-6 Conjugate-Gradient Method
4-7 Steepest-Descent Method
4-8 Comparative Performance Test
4-9 Green's Functions: Spectral and Continued-Fraction Representations 
 

5 GENETIC CODE OF SPECTRAL DENSITIES

5-1 Finite Delta_k-Sequences
5-2 Spectral Densities with Bounded Support
5-3 Spectral Densities with Bounded Support and a Gap
5-4 Spectral Densities with Unbounded Support
5-5 Spectral Densities with Unbounded Support and a Gap
5-6 Orthogonal Polynomials
 

6 RECURSION METHOD ILLUSTRATED 

6-1 Harmonic Oscillator
6-2 Spin Waves
6-3 Lattice Fermions
6-4 Quantum Spins
6-5 Classical Spins 
 

7 UNIVERSALITY CLASSES OF DYNAMICAL BEHAVIOR 

7-1 Dynamics of the Equivalent-Neighbor XYZ Model
7-2 Fluctuation Functions and Spectral Densities for the XXZ  Case
7-3 Recursion Method Applied to Equivalent-Neighbor Spin Models
7-4 Quantum Equivalent-Neighbor XYZ Model
7-5 Prototype Universality Classes
7-6 Two-Sublattice Spin Model with Long-Range Interaction
7-7 Many-Body Systems with Short-Range Interaction 
 

8 TERMINATION OF CONTINUED FRACTIONS:
    ATTEMPTS AT DAMAGE CONTROL 

8-1 Cut-Off Termination
8-2 n-Pole Approximation
8-3 Pole Locations and Spectral Densities 
8-4 Memory Functions and Fluctuation Functions
8-5 Moving Beyond Truncation 
 

9 RECONSTRUCTION OF SPECTRAL DENSITIES FROM
    INCOMPLETE CONTINUED FRACTIONS 

9-1 Model Terminators from Model Spectral Densities
9-2 Square-Root Terminator
9-3 Rectangular Terminator
9-4 Endpoint Singularities
9-5 Beta-Terminator
9-6 Gap Terminators
9-7 Infrared Singularities in Spectral Densities with Bounded Support
9-8 Spectral Densities with a -Function Central Peak
9-9 Terminator with Matching Infrared Singularity
9-10 Compact  Alpha-Terminator
9-11 Gaussian Terminator
9-12 Infrared Singularities in Spectral Densities with Unbounded Support
9-13 Unbounded  Alpha-Terminator
9-14 Split-Gaussian Terminator 
 

10 TRANSPORT OF SPIN FLUCTUATIONS AT HIGH TEMPERATURE 

10-1 Generic High-Temperature Spin Dynamics
10-2 1D s=1/2 XYZ Model on Semi-Infinite Chain
10-3 Spin-1/2 XX Model: Neither Spin Diffusion nor Exponential Relaxation
10-4 Boundary Effects: Buildup of an Infrared Divergence
10-5 Boundary Effects: Crossover Between Growth Rates
10-6 Spin-1/2 XXZ Model
10-7 From Gaussian Decay to Exponential Decay
10-8 Analysis of Delta_k-Sequences with Growth Rates near  lambda=1
10-9 From Exponential Relaxation to Diffusive Long-Time Tails
10-10 Sustained Power-Law Decay
10-11 From Ballistic to Diffusive Transport of Spin Fluctuations
10-12 Boundary-Spin Spectral Densities
10-13 Spectral Signature of Quantum Spin Diffusion in Dimensions d=1,2,3
10-14 Spin Diffusion in the Classical Heisenberg Model
10-15 Is Classical Spin Diffusion Anomalous?
10-16 Exponential Decay versus Long-Time Tails
10-17 Anomalous Exponent or Non-Asymptotic Effect?
10-18 Experimental Evidence for Anomalous Spin Diffusion
10-19 Two Kinds of Computational Errors
10-20 q-Dependent Correlation Function
10-21 Power Law Long-Time Tail with Logarithmic Correction
10-22 Effective Exponent
10-23 Effect of Exchange Inhomogeneities 
 

11 QUANTUM SPIN DYNAMICS AT ZERO TEMPERATURE

11-1 1D s=1/2 XY Model with Magnetic Field
11-2 Product Ground State of the 1D Spin-s XYZ Model with Magnetic Field
11-3 Conditions for the Existence of Linear Spin Waves
11-4 Resonances with Intrinsic Width
11-5 Finite and Infinite Bandwidths
11-6 Limitations of Single-Mode Picture
11-7 Spin Dynamics at Tc=0 Critical Point:
           Exact Results for the Transverse Ising Model and the XX Model
11-8 Long-Time Asymptotic Expansions
11-9 Structure Functions and their Singularities
11-10 Structure Functions Reconstructed by Continued-Fraction Analysis
11-11 Dynamic Structure Factors S_zz(q,omega)_TI and S_zz(q,omega)_XX:
             Two-Particle Spectrum
11-12 Dynamic Structure Factor S_xx(q,omega)_XX: Continued-Fraction Analysis
11-13 1D s=1/2 XYZ Model: Ground State and Excitation Spectrum
11-14 1D s=1/2 XXZ Model: Criticality and Long-Range Order
11-15 Excitation Spectrum of the 1D s=1/2 XXZ Ferromagnet:
             Spin Waves and Bound States
11-16 Excitation Spectrum of the 1D s=1/2 XXZ Antiferromagnet:
             Spinons and Solitons
11-17 Equal-Time Correlation Functions
11-18 Dynamic Correlation Functions
11-19 Continuum Approximation (Luttinger Model)
11-20 Hartree-Fock Approximation
11-21 Weak-Coupling and Strong-Coupling Regimes
11-22 Infrared Singularities in S_µµ(pi,omega) and S^µµ(omega)
11-23 Reconstruction of S_zz(pi,omega) (Weak-Coupling Analysis)
11-24 Reconstruction of S_xx(q,omega) (Strong-Coupling Analysis)
11-25 Strong-Coupling Reconstruction of S^xx(omega)
11-26 2D s=1/2 XXZ Antiferromagnet
11-27 Dynamic Structure Factors S_xx(pi,pi,omega) and S_zz(pi,pi,omega)
11-28 1D Spin-1 Heisenberg Antiferromagnet with Uniaxial
              Exchange and Single-Site Anisotropy
11-29 Dynamically Relevant Excitation Gaps
11-30 Dispersion and Line Shapes
 

BIBLIOGRAPHY