Spectral properties for Schrödinger operators are a major concern in quantum mechanics both in physics and in mathematics. For the few-particle systems, we now have sufficient knowledge for two-body systems, although much less is known about N-body systems. The asymptotic completeness of time-dependent wave operators was proved in the 1980s and was a landmark in the study of the N-body problem. However, many problems are left open for the stationary N-particle equation. Due to the recent rapid development of computer power, it is now possible to compute the three-body scattering problem numerically, in which the stationary formulation of scattering is used. This means that the stationary theory for N-body Schrödinger operators remains an important problem of quantum mechanics. It is stressed here that for the three-body problem, we have a satisfactory stationary theory. This book is devoted to the mathematical aspects of the N-body problem from both the time-dependent and stationary viewpoints. The main themes are:
(1) The Mourre theory for the resolvent of self-adjoint operators
(2) Two-body Schrödinger operators—Time-dependent approach and stationary approach
(3) Time-dependent approach to N-body Schrödinger operators
(4) Eigenfunction expansion theory for three-body Schrödinger operators
Compared with existing books for the many-body problem, the salient feature of this book consists in the stationary scattering theory (4). The eigenfunction expansion theorem is the physical basis of Schrödinger operators. Recently, it proved to be the basis of inverse problems of quantum scattering. This book provides necessary background information to understand the physical and mathematical basis of Schrödinger operators and standard knowledge for future development.
Author(s): Hiroshi Isozaki
Series: Mathematical Physics Studies
Edition: 1
Publisher: Springer Nature Singapore
Year: 2023
Language: English
Pages: 399
Tags: Quantum Mechanics, Schrödinger Operators, Many-Body Systems
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
