Co-groups, commutator methods and spectral theory

The conjugate operator method is a powerful recently developed technique for studying spectral properties of self-adjoint operators. One of the purposes of this volume is to present a refinement of the original method due to Mourre leading to essentially optimal results in situations as varied as ordinary differential operators, pseudo-differential operators and N -body Schrdinger hamiltonians. Another topic is a new algebraic framework for the N -body problem allowing a simple and systematic treatment of large classes of many-channel hamiltonians. The monograph will be of interest to research mathematicians and mathematical physicists. The authors have made efforts to produce an essentially self-contained text, which makes it accessible to advanced students. Thus about one third of the book is devoted to the development of tools from functional analysis, in particular real interpolation theory for Banach spaces and functional calculus and Besov spaces associated with multi-parameter C o-groups. Table of Contents: Preface Comments on notations Chapter 1. Some Spaces of Functions and Distributions 1.1. Calculus on Euclidean Spaces 1.2. Distributions, Fourier Transforms 1.3. Estimates of Functions and their Fourier Transforms 1.4. Rapidly Decreasing Distributions Chapter 2. Real Interpolation of Banach Spaces 2.1. Banach Spaces and Linear Operators 2.2. The K -Functional 2.3. The Mean and the Trace Method 2.4. Comparison and Duality of Interpolation Spaces 2.5. The Reiteration Theorem 2.6. Interpolation of Operators 2.7. Quasi-Linearizable Couples of B -Spaces 2.8. Friedrichs Couples Chapter 3. C 0-Groups and Functional Calculi 3.1. Submultiplicative Functions and Algebras Associated to them 3.2. C 0-Groups: Continuity Properties and Elementary Functional Calculus 3.3. The Discrete Sobolev Scale Associated to a C 0-Group 3.4. Besov Spaces Associated to a C 0-Group 3.5. Littlewood-Paley Estimates 3.6. Polynomially Bounded C 0-Groups 3.7. C 0-Groups in Hilbert Spaces Chapter 4. Some Examples of C 0-Groups 4.1. Weighted Sobolev And Besov Spaces 4.2. C 0-Groups Associated to Vector Fields Chapter 5. Automorphisms Associated to C 0-Representations 5.1. Regularity and Commutators 5.2. Regularity of Fractional Order 5.3. Regularity Preserving and Regularity Improving Operators 5.4. The spaces r /s,p(R n ) 5.5. Commutator Expansions 5.A. Appendix: Differentiability Properties of Operator-Valued Functions Chapter 6. Unitary Representations and Regularity 6.1. Remarks on the Functional Calculus for Self-adjoint Operators 6.2. Regularity of Self-adjoint Operators with respect to UnitaryC 0-Groups 6.3. Unitary Groups in Friedrichs Couples 6.4. Estimates on (H 1)- (H 2) 6.A. Appendix: Remarks on the Functional Calculus Associated to in B ( ) Chapter 7. The Conjugate Operator Method 7.1. Locally Smooth Operators and Boundary Values Of the Resolvent 7.2. The Mourre Estimate 7.3. The Method of Differential Inequalities 7.4. Self-adjoint Operators with a Spectral Gap 7.5. Hamiltonians Associated to Symmetric Operators in Friedrichs Couples 7.6. The Limiting Absorption Principle for Some Classes of Pseudodifferential operators 7.A. Appendix: The Gronwall Lemma 7.B. Appendix: A Counterexample. Optimality of the Results on the Limiting Absorption Principle 7.C. Appendix: Asymptotic Velocity for H = h (P ) Chapter 8. An Algebraic Framework for the Many-Body Problem 8.1. Self-adjoint Operators Affiliated to C -Algebras 8.2. Tensor products 8.3. -Functions in a C -Algebra Setting 8.4. Graded C -Algebras Chapter 9. Spectral theory of N -Body Hamiltonians 9.1. Tensorial Factorization of (X ) 9.2. Semicompact Operators 9.3. The N -Body Algebra 9.4. Non-Relativistic N -Body Hamiltonians 9.A. Appendix: Remarks on the 1,1 Property Chapter 10. Quantum-Mechanical N -Body Systems 10.1. Clustering of Particles 10.2. Quantum-Mechanical N -Body Hamiltonians Bibliography Notations Index Series: Progress in Mathematics, Volume 135