A Method for Studying Model Hamiltonians

A Method for Studying Model Hamiltonians: A Minimax Principle for Problems in Statistical Physics centers on methods for solving certain problems in statistical physics which contain four-fermion interaction. Organized into four chapters, this book begins with a presentation of the proof of the asymptotic relations for the many-time correlation functions. Chapter 2 details the construction of a proof of the generalized asymptotic relations for the many-time correlation averages. Chapter 3 explains the correlation functions for systems with four-fermion negative interaction. The last chapter shows the model systems with positive and negative interaction components.

Series Editor's Preface

Preface

Introduction

§ 1. General Remarks

§ 2. Remarks an Quasi-Averages

Chapter 1. Proof of the Asymptotic Relations for the Many-Time Correlation Functions

§ 1. General Treatment of the Problem. Some Preliminary Results and Formulation of the Problem

§ 2. Equations of Motion and Auxiliary Operator Inequalities

§ 3. Additional Inequalities

§ 4. Bounds for the Difference of the Single-Time Averages

§ 5. Remark (I)

§ 6. Proof of the Closeness of Averages Constructed on the Basis of Model and Trial Hamiltonians for "Normal" Ordering of the Operators in the Averages

§ 7. Proof of the Closeness of the Averages for Arbitrary Ordering of the Operators in the Averages

Remark (II)

§ 8. Estimates of the Asymptotic Closeness of the Many-Time Correlation Averages

Chapter 2. Construction of a Proof of the Generalized Asymptotic Relations for the Many-Time Correlation Averages

§ 1. Selection Rules and Calculation of the Averages

§ 2. Generalized Convergence

§ 3. Remark

§ 4. Proof of the Asymptotic Relations

§ 5. Remark on the Construction of Uniform Bounds

§ 6. Generalized Asymptotic Relations for the Green's Functions

§ 7. The Existence of Generalized Limits

Chapter 3. Correlation Functions for Systems with Four-Fermion Negative Interaction

§ 1. Calculation of the Free Energy for Model Systems with Attraction

§ 2. Further Properties of the Expressions for the Free Energy

§ 3. Construction of Asymptotic Relations for the Free Energy

§ 4. On the Uniform Convergence with Respect to θ of the Free Energy Function and on Bounds for the Quantities δv

§ 5. Properties of Partial Derivatives of the Free Energy Function. Theorem 3.III

§ 6. Rider to Theorem 3.III and Construction of an Auxiliary Inequality

§ 7. On the Difficulties of Introducing Quasi-Averages

§ 8. A New Method of Introducing Quasi-Averages

§ 9. The Question of the Choice of Sign for the Source-Terms

§ 10. The Construction of Upper-Bound Inequalities in the Case When C=0

Chapter 4. Model Systems with Positive and Negative Interaction Components

§ 1. Hamiltonian with Negative Coupling Constants (Repulsive Interaction)

§ 2. Features of the Asymptotic Relations for the Free Energies in the Case of Systems with Positive Interaction

§ 3. Bounds for the Free Energies and Correlation Functions

§ 4. Examination of an Auxiliary Problem

§ 5. Solution of the Question of Uniqueness

§ 6. Hamiltonians with Coupling Constants of Different Signs. The Minimax Principle

References

Index