Many-Body Schrödinger Equation - Scattering Theory and Eigenfunction Expansions

Preface to the Japanese Edition
	References
Preface to the English Edition
Contents
1 Self-adjoint Operators and Spectra
	1.1 Introduction
	1.2 Self-adjointness
		1.2.1 Self-adjoint Operator
		1.2.2 Example
		1.2.3 Condition for the Self-adjointness
		1.2.4 Essential Self-adjointness
		1.2.5 Perturbation for Self-adjoint Operators
		1.2.6 Example
		1.2.7 Relative Compactness
	1.3 Spectral Decomposition
		1.3.1 Resolvent and Spectra
		1.3.2 Example
		1.3.3 Spectral Decomposition
		1.3.4 Example
		1.3.5 The Helffer–Sjöstrand Formula
		1.3.6 Supplement to the Kato–Rellich Theorem
	1.4 Classification of Spectra
		1.4.1 Discrete Spectrum and Essential Spectrum
		1.4.2 Point Spectrum and Continuous Spectrum
		1.4.3 Absolutely Continuous Spectrum and Singular Continuous Spectrum
	1.5 Bound States and Scattering States
	1.6 Supplements to Self-adjointness
		1.6.1 Quadratic Forms
		1.6.2 Nelson\'s Commutator Theorem
		1.6.3 Generator of Dilation Group
	1.7 Mourre Theory
		1.7.1 Main Results
		1.7.2 Proof of Theorem 1.49 (I)
		1.7.3 Proof of Theorem 1.49 (II)
		1.7.4 Proof of Theorem 1.49 (III)
	1.8 Applications to - Δ
	References
2 Two-Body Problem
	2.1 Introduction
	2.2 Improvement of Mourre Theory
		2.2.1 A Problem in Mourre Theory
		2.2.2 The Mourre Inequality
		2.2.3 Limiting Absorption Principle
		2.2.4 Two-Body Schrödinger Operators
	2.3 Smooth Perturbations
		2.3.1 Smooth Perturbation Theory
		2.3.2 Decay of e-itH
	2.4 Enss Method
	2.5 Heisenberg Form and Propagation of Wave Packets
		2.5.1 Commutation Relations and Propagation of Wave Packets
		2.5.2 Estimates of Heisenberg Derivatives
		2.5.3 High-Velocity Estimates
		2.5.4 Estimates for the Main Part
		2.5.5 Low-Velocity Estimates
	References
3 Asymptotic Completeness  for Many-Body Systems
	3.1 Introduction
	3.2 Hamlitonians for Atoms
		3.2.1 Atoms with Many Electrons
		3.2.2 Tensor Product
		3.2.3 Cluster Decomposition and Spectrum
		3.2.4 Discrete Spectrum
	3.3 N-Body Schrödinger Operators
		3.3.1 Separation of Center of Mass
		3.3.2 Jacobi Coordinates
		3.3.3 Definition of N-Body Hamiltonian
		3.3.4 Cluster Decomposition
		3.3.5 Examples
		3.3.6 Intercluster Subspaces
		3.3.7 Dual Space and Jacobi Coordinates
	3.4 Essential Spectrum
		3.4.1 Partition of Unity
		3.4.2 Theorem of Zhislin, van Winter, Hunziker
	3.5 Mourre Inequality
		3.5.1 Thresholds
		3.5.2 Mourre Inequality
	3.6 Convex Function and Commutator
		3.6.1 Partition of mathcalX
		3.6.2 Construction of the Convex Function
		3.6.3 Properties of Partition of Unity {ga(x)}
	3.7 Propagation of Wave Packets
		3.7.1 High-Velocity Estimates
		3.7.2 Estimates for the Main Part
		3.7.3 Low-Velocity Estimates
		3.7.4 Square Root and Commutator
	3.8 Asymptotic Completeness
	References
4 Resolvent of Multi-particle System
	4.1 Introduction
	4.2 Limiting Absorption Principle
	4.3 Decay of L2-Eigenfunctions
	4.4 Algebra of Operators and Commutators
		4.4.1 Asymptotic Expansion of Commutators
		4.4.2 Algebra of Operators
		4.4.3 The Space with Weight langleArangle
		4.4.4 The Choice of A
		4.4.5 Examples of Elements of mathcalOP0,2(A)
		4.4.6 Operator B
		4.4.7 Algebra Using B
	4.5 Sommerfeld–Rellich Type Uniqueness Theorem
	4.6 Generalization of the Radiation Condition
	4.7 Micro-local Resolvent Estimates
		4.7.1 Estimates by Commutators
		4.7.2 Improvement of the Estimate
	4.8 Applications
		4.8.1 Differentiability with Respect to the Energy Parameter
		4.8.2 Localization by Pseudo-Differential Operators
		4.8.3 Resolvent Estimates in the Conic Region
		4.8.4 Decay of the Unitary Group
	4.9 Yafaev\'s Function
	4.10 Precise Radiation Condition
		4.10.1 Yafaev\'s Estimate
		4.10.2 Reformulation by ΨDO
	References
5 Three-Body Problem and the Eigenfunction Expansion
	5.1 Introduction
		5.1.1 Helmholtz Equation
		5.1.2 Generalization of Fourier Transformation
		5.1.3 The Three-Body Problem and the Eigenoperator Expansion
	5.2 mathcalB-mathcalBast Spaces
	5.3 Restriction of the Fourier Transform on the Sphere
	5.4 The Fourier Transform Associated with Ha
		5.4.1 Definition of the Fourier Transform
		5.4.2 Asymptotic Behavior of the Resolvent and the Fourier Transform
	5.5 Asymptotic Expansion of the Resolvent of Ha
	5.6 Wave Operator
	5.7 Eigenoperator Expansion
	5.8 Asymptotic Behavior of Solutions to the Stationary Three-Body Schrödinger Equation
		5.8.1 Asymptotic Expansion of the Resolvent
		5.8.2 S-Matrix
		5.8.3 Three-Body Helmholtz Equation
	5.9 The Structure of the S-Matrix
		5.9.1 Collision Process
		5.9.2 Generalized Eigenfunctions
		5.9.3 Analytic Continuation
		5.9.4 Scattering Cross-Section
	References
6 Supplement
	6.1 Fourier Transform
	6.2 Interpolation Theorem
	6.3 Pseudo-differential Operators
	6.4 Almost Analytic Extension
	6.5 Lebesgue Decomposition of Measure
		6.5.1 Borel σ-Field
		6.5.2 Lebesgue–Stieltjes Measure
		6.5.3 Absolute and Singular Continuity
	6.6 Limiting Absorption and Limiting Amplitude
		6.6.1 Limiting Absorption Principle
		6.6.2 Limiting Amplitude Principle
	6.7 Rigged Hilbert Space
	6.8 Generalized Many-Body Schrödinger Operator
	6.9 Method of Stationary Phase
	6.10 Proof of Theorem 4.16
	6.11 Flow Along Characteristic Curve
	6.12 Abstract Radiation Condition
	6.13 Abstract Stationary Theory for the Two-Body Problem
		6.13.1 Stationary Wave Operators
		6.13.2 Time-Dependent Wave Operators
		6.13.3 Spectral Representation
		6.13.4 S-Matrix
		6.13.5 Application to Two-Body Schrödinger Operators
	6.14 Long-Range Scattering
	6.15 Proof of Theorem 5.1
	6.16 Low-Energy Behavior of the Resolvent
		6.16.1 Zero-Resonance
		6.16.2 Asymptotic Expansion of the Resolvent
		6.16.3 L2-Eigenfunctions
		6.16.4 Number of Eigenvalues
	6.17 The Proof of (5.72) in Sect. 5.4.2
	6.18 Pointwise Estimate of the Resolvent
	References
Appendix  Related Literature
Notes Added to the English Translation
	References
Index