Wave Packet Analysis of Feynman Path Integrals

Fabio Nicola, S. Ivan Trapasso (Autoren)

Buch | Softcover

XIII, 214 Seiten

2022 | 1st ed. 2022
Springer International Publishing (Verlag)
978-3-031-06185-1 (ISBN)

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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.

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.

- 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)

Erscheinungsdatum	30.07.2022
Reihe/Serie	Lecture Notes in Mathematics
Zusatzinfo	XIII, 214 p. 3 illus. in color.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	355 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
	Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
	Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
Schlagworte	Coherent states • Feynman path integrals • Gabor Wave Packets • Harmonic Analysis • Pointwise Convergence of Kernels • pseudodifferential operators • Schroedinger equation • time-frequency analysis • Time Slicing Approximation
ISBN-10	3-031-06185-3 / 3031061853
ISBN-13	978-3-031-06185-1 / 9783031061851
Zustand	Neuware