Wave packet analysis of Feynman path integrals / / Fabio Nicola, S. Ivan Trapasso |
Autore | Nicola Fabio |
Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2022] |
Descrizione fisica | 1 online resource (220 pages) |
Disciplina | 515.43 |
Collana | Lecture notes in mathematics |
Soggetto topico |
Feynman integrals
Gabor transforms Quantum theory Integrals de Feynman Teoria quàntica |
Soggetto genere / forma | Llibres electrònics |
ISBN | 3-031-06186-1 |
Formato | Materiale a stampa |
Livello bibliografico | Monografia |
Lingua di pubblicazione | eng |
Nota di contenuto |
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 -- 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. |
Record Nr. | UNISA-996483172503316 |
Nicola Fabio | ||
Cham, Switzerland : , : Springer, , [2022] | ||
Materiale a stampa | ||
Lo trovi qui: Univ. di Salerno | ||
|
Wave packet analysis of Feynman path integrals / / Fabio Nicola, S. Ivan Trapasso |
Autore | Nicola Fabio |
Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2022] |
Descrizione fisica | 1 online resource (220 pages) |
Disciplina | 515.43 |
Collana | Lecture notes in mathematics |
Soggetto topico |
Feynman integrals
Gabor transforms Quantum theory Integrals de Feynman Teoria quàntica |
Soggetto genere / forma | Llibres electrònics |
ISBN | 3-031-06186-1 |
Formato | Materiale a stampa |
Livello bibliografico | Monografia |
Lingua di pubblicazione | eng |
Nota di contenuto |
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 -- 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. |
Record Nr. | UNINA-9910585774903321 |
Nicola Fabio | ||
Cham, Switzerland : , : Springer, , [2022] | ||
Materiale a stampa | ||
Lo trovi qui: Univ. Federico II | ||
|