Mathematical Foundations of Quantum Statistical Mechanics

1. Evolution of States of Quantum Systems of Finitely Many Particles.- 1. Principal Concepts of Quantum Mechanics.- 2. Evolution of States of Quantum Systems with Arbitrarily Many Particles.- 3. Evolution of States in the Heisenberg Representation and in the Interaction Representation.- Mathematical Supplement I.- References.- 2. Evolution of States of Infinite Quantum Systems.- 4. Bogolyubov Equations for Statistical Operators.- 5. Solution of the Bogolyubov Equations.- 6. Gibbs Distributions.- Mathematical Supplement II.- Mathematical Supplement III.- 3. Thermodynamic Limit.- 7. Thermodynamic Limit for Statistical Operators.- 8. Statistical Operators in the Case of Quantum Statistics.- 9. Bogolyubov’s Principle of Weakening of Correlations.- Mathematical Supplement IV.- References.- 4. Mathematical Problems in the Theory of Superconductivity.- 10. Fröhlich Model.- 11. Bogolyubov’s Compensation Principle for “Dangerous” Diagrams. Compensation Equations.- 12. Bardeen-Cooper-Schrieffer (BCS) Hamiltonian.- 13. Microscopic Theory of Superfluidity.- Mathematical Supplement V.- 5. Green’s Functions.- 14. Green’s Functions. Equations for Green’s Functions.- 15. Investigation of the Equations for Green’s Functions in the Theory of Superconductivity and Superfluidity.- 16. Green’s Functions in the Thermodynamic Limit.- 6. Exactly Solvable Models.- 17. Description of the Hamiltonians of Model Systems.- 18. Functional Spaces of Translation-Invariant Functions.- 19. Model Hamiltonians in the Spaces of Translation Invariant Functions.- 20. Model BCS Hamiltonian in the Space hT. Equivalence of General and Model Hamiltonians in the Space of Pairs.- 21. Equations for Green’s Functions and Their Solutions.- Mathematical Supplement VI.- 7. Quasiaverages. Theorem onSingularities of Green’s Functions of 1/q2 -Type.- 22. Quasiaverages.- 23. Green’s Functions and Their Spectral Representations.- 24. Theorem on Singularities of Green’s Functions of 1/q2 -Type.