Full Description
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.
Contents
- 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.
