Contents Preface xi Basic Notation 1 Introduction 5 Chapter 0. Basic Concepts 17 1. Classification of the spectrum 17 2. Classes of compact operators 20 3. The resolvent equation. Conditions for self-adjointness 23 4. Wave operators (WO) 26 5. The smooth method 29 6. The stationary scheme 33 7. The scattering operator and the scattering matrix (SM) 38 8. The trace class method 42 9. The spectral shift function (SSF) and the perturbation determinant (PD) 45 10. Differential operators 52 11. Function spaces and embedding theorems 56 12. Pseudodifferential operators 58 13. Miscellaneous analytic facts 67 Chapter 1. Smooth Theory. The Schr¨ odinger Operator 71 1. Trace theorems 71 2. The free Hamiltonian 75 3. The Schr¨ odinger operator 79 4. Existence of wave operators 82 5. Wave operators for long-range potentials 86 6. Completeness of wave operators 93 7. The limiting absorption principle (LAP) 95 8. The scattering matrix 96 9. Absence of the singular continuous spectrum 98 10. General differential operators of second order 101 11. The perturbed polyharmonic operator 103 12. The Pauli and Dirac operators 104 Chapter 2. Smooth Theory. General Differential Operators 109 1. Spectral analysis of differential operators with constant coefficients 109 2. Scalar differential operators 116 3. Nonelliptic differential operators 118 4. Matrix differential operators 122 vii
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