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Table of Contents
Intro
Preface
Contents
Outline
1 Itinerary: How Gabor Analysis Met Feynman Path Integrals
1.1 The Elements of Gabor Analysis
1.1.1 The Analysis of Functions via Gabor Wave Packets
1.2 The Analysis of Operators via Gabor Wave Packets
1.2.1 The Problem of Quantization
1.2.2 Metaplectic Operators
1.3 The Problem of Feynman Path Integrals
1.3.1 Rigorous Time-Slicing Approximation of Feynman Path Integrals
1.3.2 Pointwise Convergence at the Level of Integral Kernels for Feynman-Trotter Parametrices
1.3.3 Convergence of Time-Slicing Approximations in L(L2) for Low-Regular Potentials
1.3.4 Convergence of Time-Slicing Approximations in the Lp Setting
Part I Elements of Gabor Analysis
2 Basic Facts of Classical Analysis
2.1 General Notation
2.2 Function Spaces
2.2.1 Lebesgue Spaces
2.2.2 Differentiable Functions and Distributions
2.3 Basic Operations on Functions and Distributions
2.4 The Fourier Transform
2.4.1 Convolution and Fourier Multipliers
2.5 Some More Facts and Notations
3 The Gabor Analysis of Functions
3.1 Time-Frequency Representations
3.1.1 The Short-Time Fourier Transform
3.1.2 Quadratic Representations
3.2 Modulation Spaces
3.3 Wiener Amalgam Spaces
3.4 A Banach-Gelfand Triple of Modulation Spaces
3.5 The Sjöstrand Class and Related Spaces
3.6 Complements
3.6.1 Weight Functions
3.6.2 The Cohen Class of Time-Frequency Representations
3.6.3 Kato-Sobolev Spaces
3.6.4 Fourier Multipliers
3.6.5 More on the Sjöstrand Class
3.6.6 Boundedness of Time-Frequency Transforms on Modulation Spaces
3.6.7 Gabor Frames
4 The Gabor Analysis of Operators
4.1 The General Program
4.2 The Weyl Quantization
4.3 Metaplectic Operators
4.3.1 Notable Facts on Symplectic Matrices
4.3.2 Metaplectic Operators: Definitions and Basic Properties
4.3.3 The Schrödinger Equation with Quadratic Hamiltonian
4.3.4 Symplectic Covariance of the Weyl Calculus
4.3.5 Gabor Matrix of Metaplectic Operators
4.4 Fourier and Oscillatory Integral Operators
4.4.1 Canonical Transformations and the Associated Operators
4.4.2 Generalized Metaplectic Operators
4.4.3 Oscillatory Integral Operators with Rough Amplitude
4.5 Complements
4.5.1 Weyl Operators and Narrow Convergence
4.5.2 General Quantization Rules
4.5.3 The Class FIO'(S,vs)
4.5.4 Finer Aspects of Gabor Wave Packet Analysis
5 Semiclassical Gabor Analysis
5.1 Semiclassical Transforms and Function Spaces
5.1.1 Sobolev Spaces and Embeddings
5.2 Semiclassical Quantization, Metaplectic Operators and FIOs
Part II Analysis of Feynman Path Integrals
6 Pointwise Convergence of the Integral Kernels
6.1 Summary
6.2 Preliminary Results
6.2.1 The Schwartz Kernel Theorem
6.2.2 Uniform Estimates for Linear Changes of Variable
Preface
Contents
Outline
1 Itinerary: How Gabor Analysis Met Feynman Path Integrals
1.1 The Elements of Gabor Analysis
1.1.1 The Analysis of Functions via Gabor Wave Packets
1.2 The Analysis of Operators via Gabor Wave Packets
1.2.1 The Problem of Quantization
1.2.2 Metaplectic Operators
1.3 The Problem of Feynman Path Integrals
1.3.1 Rigorous Time-Slicing Approximation of Feynman Path Integrals
1.3.2 Pointwise Convergence at the Level of Integral Kernels for Feynman-Trotter Parametrices
1.3.3 Convergence of Time-Slicing Approximations in L(L2) for Low-Regular Potentials
1.3.4 Convergence of Time-Slicing Approximations in the Lp Setting
Part I Elements of Gabor Analysis
2 Basic Facts of Classical Analysis
2.1 General Notation
2.2 Function Spaces
2.2.1 Lebesgue Spaces
2.2.2 Differentiable Functions and Distributions
2.3 Basic Operations on Functions and Distributions
2.4 The Fourier Transform
2.4.1 Convolution and Fourier Multipliers
2.5 Some More Facts and Notations
3 The Gabor Analysis of Functions
3.1 Time-Frequency Representations
3.1.1 The Short-Time Fourier Transform
3.1.2 Quadratic Representations
3.2 Modulation Spaces
3.3 Wiener Amalgam Spaces
3.4 A Banach-Gelfand Triple of Modulation Spaces
3.5 The Sjöstrand Class and Related Spaces
3.6 Complements
3.6.1 Weight Functions
3.6.2 The Cohen Class of Time-Frequency Representations
3.6.3 Kato-Sobolev Spaces
3.6.4 Fourier Multipliers
3.6.5 More on the Sjöstrand Class
3.6.6 Boundedness of Time-Frequency Transforms on Modulation Spaces
3.6.7 Gabor Frames
4 The Gabor Analysis of Operators
4.1 The General Program
4.2 The Weyl Quantization
4.3 Metaplectic Operators
4.3.1 Notable Facts on Symplectic Matrices
4.3.2 Metaplectic Operators: Definitions and Basic Properties
4.3.3 The Schrödinger Equation with Quadratic Hamiltonian
4.3.4 Symplectic Covariance of the Weyl Calculus
4.3.5 Gabor Matrix of Metaplectic Operators
4.4 Fourier and Oscillatory Integral Operators
4.4.1 Canonical Transformations and the Associated Operators
4.4.2 Generalized Metaplectic Operators
4.4.3 Oscillatory Integral Operators with Rough Amplitude
4.5 Complements
4.5.1 Weyl Operators and Narrow Convergence
4.5.2 General Quantization Rules
4.5.3 The Class FIO'(S,vs)
4.5.4 Finer Aspects of Gabor Wave Packet Analysis
5 Semiclassical Gabor Analysis
5.1 Semiclassical Transforms and Function Spaces
5.1.1 Sobolev Spaces and Embeddings
5.2 Semiclassical Quantization, Metaplectic Operators and FIOs
Part II Analysis of Feynman Path Integrals
6 Pointwise Convergence of the Integral Kernels
6.1 Summary
6.2 Preliminary Results
6.2.1 The Schwartz Kernel Theorem
6.2.2 Uniform Estimates for Linear Changes of Variable