The purpose of this monograph is to offer an accessible and essentially self-contained presentation of some mathematical aspects of the Feynman path integral in non-relativistic quantum mechanics. In spite of the primary role in the advancement of modern theoretical physics and the wide range of applications, path integrals are still a source of challenging problem for mathematicians. From this viewpoint, path integrals can be roughly described in terms of approximation formulas for an operator (usually the propagator of a Schrödinger-type evolution equation) involving a suitably designed sequence of operators.
In keeping with the spirit of harmonic analysis, the guiding theme of the book is to illustrate how the powerful techniques of time-frequency analysis - based on the decomposition of functions and operators in terms of the so-called Gabor wave packets – can be successfully applied to mathematical path integrals, leading to remarkable results and paving the way to a fruitful interaction.
This monograph intends to build a bridge between the communities of people working in time-frequency analysis and mathematical/theoretical physics, and to provide an exposition of the present novel approach along with its basic toolkit. Having in mind a researcher or a Ph.D. student as reader, we collected in Part I the necessary background, in the most suitable form for our purposes, following a smooth pedagogical pattern. Then Part II covers the analysis of path integrals, reflecting the topics addressed in the research activity of the authors in the last years.
Author(s): Fabio Nicola, S. Ivan Trapasso
Series: Lecture Notes in Mathematics, 2305
Publisher: Springer
Year: 2022
Language: English
Pages: 219
City: Cham
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
6.2.3 Exponentiation in Banach Algebras
6.2.4 Two Technical Lemmas
6.3 Reduction to the Case .12em.1emdotteddotteddotted.76dotted.6h=(2π)-1
6.4 The Fundamental Solution and the Trotter Formula
6.5 Potentials in M∞0,s
6.6 Potentials in C∞b
6.7 Potentials in the Sjöstrand Class M∞,1
6.8 Convergence at Exceptional Times
6.9 Physics at Exceptional Times
7 Convergence in L(L2) for Potentials in the Sjöstrand Class
7.1 Summary
7.2 An Abstract Approximation Result in L(L2)
7.3 Short-Time Analysis of the Action
7.4 Estimates for the Parametrix and Convergence Results
8 Convergence in L(L2) for Potentials in Kato-Sobolev Spaces
8.1 Summary
8.2 Sobolev Regularity of the Hamiltonian Flow
8.3 Sobolev Regularity of the Classical Action
8.4 Analysis of the Parametrices and Convergence Results
8.5 Higher-Order Parametrices
9 Convergence in the Lp Setting
9.1 Summary
9.2 Review of the Short Time Analysis in the Smooth Category
9.3 Wave Packet Analysis of the Schrödinger Flow
9.4 Convergence in Lp with Loss of Derivatives
9.5 The Case of Magnetic Fields
9.6 Sharpness of the Results
9.7 Extensions to the Case of Rough Potentials
Bibliography
Index
