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 wayto 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.
- 1. Itinerary: How Gabor Analysis Met Feynman Path Integrals. -
Part I Elements of Gabor Analysis. - 2. Basic Facts of Classical Analysis. - 3. The Gabor Analysis of Functions. - 4. The Gabor Analysis of Operators. - 5. Semiclassical Gabor Analysis. -
Part II Analysis of Feynman Path Integrals. - 6. Pointwise Convergence of the Integral Kernels. - 7. Convergence in
L(L2) for Potentials in the Sjöstrand Class. - 8. Convergence in
L(L2) for Potentials in Kato-Sobolev Spaces. - 9. Convergence in the
Lp Setting.
"The book is written in a reader-friendly manner, and is addressed to students and researchers interested in mathematical physics and/or time-frequency analysis. ... They have done their best to make the content accessible to a wide audience. The book can be used both for self-study and for a special course addressed to Ph.D. and masters students." (Yana Kinderknecht, Mathematical Reviews, Issue 4, March, 2024)
Fabio Nicola is Full Professor of Mathematical Analysis at Politecnico di Torino, Italy. He has authored about 100 research articles and a monograph on several topics in partial differential equations, operator theory and Fourier analysis.
S. Ivan Trapasso is a postdoctoral research fellow at the University of Genova, Italy. His main research interests are in the area of modern Fourier analysis, with applications to problems in mathematical physics and machine learning.
Über den Autor
Fabio Nicola is Full Professor of Mathematical Analysis at Politecnico di Torino, Italy. He has authored about 100 research articles and a monograph on several topics in partial differential equations, operator theory and Fourier analysis.
S. Ivan Trapasso is a postdoctoral research fellow at the University of Genova, Italy. His main research interests are in the area of modern Fourier analysis, with applications to problems in mathematical physics and machine learning.
Inhaltsverzeichnis
- Itinerary - How Gabor Analysis met Feynman Path Integrals.
- Part I Elements of Gabor Analysis.
- Basic Facts of Classical Analysis.
- The Gabor Analysis of Functions.
- The Gabor Analysis of Operators.
- Semiclassical Gabor Analysis.
- Part II Analysis of Feynman Path Integrals.
- Pointwise Convergence of the Integral Kernels.
- Convergence in L(L2) - Potentials in the Sjöstrand Class.
- Convergence in L(L2) - Potentials in Kato-Sobolev Spaces.
- Convergence in the Lp Setting.
Klappentext
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 wayto 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.
Includes a self-contained treatment of the background toolkit
Describes a novel approach to the analysis of Feynman path integrals
Provides a detailed exposition of recent advances in mathematical path integrals
