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001448473 019__ $$a1337945871
001448473 020__ $$a9783031061868$$q(electronic bk.)
001448473 020__ $$a3031061861$$q(electronic bk.)
001448473 020__ $$z3031061853
001448473 020__ $$z9783031061851
001448473 0247_ $$a10.1007/978-3-031-06186-8$$2doi
001448473 035__ $$aSP(OCoLC)1337856497
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001448473 049__ $$aISEA
001448473 050_4 $$aQC174.17.F45
001448473 08204 $$a530.12$$223/eng/20220805
001448473 1001_ $$aNicola, Fabio.
001448473 24510 $$aWave packet analysis of Feynman path integrals/$$cFabio Nicola.
001448473 260__ $$aCham, Switzerland :$$bSpringer,$$c2022.
001448473 300__ $$a1 online resource
001448473 4901_ $$aLecture notes in mathematics ;$$vv. 2305
001448473 504__ $$aIncludes bibliographical references and index.
001448473 5050_ $$aIntro -- Preface -- Contents -- Outline -- 1 Itinerary: How Gabor Analysis Met Feynman Path Integrals -- 1.1 The Elements of Gabor Analysis -- 1.1.1 The Analysis of Functions via Gabor Wave Packets -- 1.2 The Analysis of Operators via Gabor Wave Packets -- 1.2.1 The Problem of Quantization -- 1.2.2 Metaplectic Operators -- 1.3 The Problem of Feynman Path Integrals -- 1.3.1 Rigorous Time-Slicing Approximation of Feynman Path Integrals -- 1.3.2 Pointwise Convergence at the Level of Integral Kernels for Feynman-Trotter Parametrices
001448473 5058_ $$a1.3.3 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
001448473 5058_ $$a3.1.1 The Short-Time Fourier Transform -- 3.1.2 Quadratic Representations -- 3.2 Modulation Spaces -- 3.3 Wiener Amalgam Spaces -- 3.4 A Banach-Gelfand Triple of Modulation Spaces -- 3.5 The Sjöstrand Class and Related Spaces -- 3.6 Complements -- 3.6.1 Weight Functions -- 3.6.2 The Cohen Class of Time-Frequency Representations -- 3.6.3 Kato-Sobolev Spaces -- 3.6.4 Fourier Multipliers -- 3.6.5 More on the Sjöstrand Class -- 3.6.6 Boundedness of Time-Frequency Transforms on Modulation Spaces -- 3.6.7 Gabor Frames -- 4 The Gabor Analysis of Operators -- 4.1 The General Program
001448473 5058_ $$a4.2 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
001448473 5058_ $$a4.5.1 Weyl Operators and Narrow Convergence -- 4.5.2 General Quantization Rules -- 4.5.3 The Class FIO'(S,vs) -- 4.5.4 Finer Aspects of Gabor Wave Packet Analysis -- 5 Semiclassical Gabor Analysis -- 5.1 Semiclassical Transforms and Function Spaces -- 5.1.1 Sobolev Spaces and Embeddings -- 5.2 Semiclassical Quantization, Metaplectic Operators and FIOs -- Part II Analysis of Feynman Path Integrals -- 6 Pointwise Convergence of the Integral Kernels -- 6.1 Summary -- 6.2 Preliminary Results -- 6.2.1 The Schwartz Kernel Theorem -- 6.2.2 Uniform Estimates for Linear Changes of Variable
001448473 506__ $$aAccess limited to authorized users.
001448473 520__ $$aThe purpose of this monograph is to offer an accessible and essentially self-contained presentation of some mathematical aspects of the Feynman path integral in non-relativistic quantum mechanics. In spite of the primary role in the advancement of modern theoretical physics and the wide range of applications, path integrals are still a source of challenging problem for mathematicians. From this viewpoint, path integrals can be roughly described in terms of approximation formulas for an operator (usually the propagator of a Schrodinger-type evolution equation) involving a suitably designed sequence of operators. In keeping with the spirit of harmonic analysis, the guiding theme of the book is to illustrate how the powerful techniques of time-frequency analysis - based on the decomposition of functions and operators in terms of the so-called Gabor wave packets can be successfully applied to mathematical path integrals, leading to remarkable results and paving the way to a fruitful interaction. This monograph intends to build a bridge between the communities of people working in time-frequency analysis and mathematical/theoretical physics, and to provide an exposition of the present novel approach along with its basic toolkit. Having in mind a researcher or a Ph.D. student as reader, we collected in Part I the necessary background, in the most suitable form for our purposes, following a smooth pedagogical pattern. Then Part II covers the analysis of path integrals, reflecting the topics addressed in the research activity of the authors in the last years.
001448473 588__ $$aOnline resource; title from PDF title page (SpringerLink, viewed August 5, 2022).
001448473 650_0 $$aQuantum theory.
001448473 650_0 $$aFeynman integrals.
001448473 650_0 $$aGabor transforms.
001448473 655_7 $$aLlibres electrònics.$$2thub
001448473 655_0 $$aElectronic books.
001448473 7001_ $$aTrapasso, S. Ivan.
001448473 77608 $$iPrint version:$$z3031061853$$z9783031061851$$w(OCoLC)1311360331
001448473 830_0 $$aLecture notes in mathematics (Springer-Verlag) ;$$v2305.
001448473 852__ $$bebk
001448473 85640 $$3Springer Nature$$uhttps://univsouthin.idm.oclc.org/login?url=https://link.springer.com/10.1007/978-3-031-06186-8$$zOnline Access$$91397441.1
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001448473 980__ $$aBIB
001448473 980__ $$aEBOOK
001448473 982__ $$aEbook
001448473 983__ $$aOnline
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