LEADER 05944nam 2200577 450 001 996483172503316 005 20230727185456.0 010 $a3-031-06186-1 035 $a(MiAaPQ)EBC7052885 035 $a(Au-PeEL)EBL7052885 035 $a(CKB)24286234000041 035 $a(PPN)263897001 035 $a(EXLCZ)9924286234000041 100 $a20230106d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aWave packet analysis of Feynman path integrals /$fFabio Nicola, S. Ivan Trapasso 210 1$aCham, Switzerland :$cSpringer,$d[2022] 210 4$d©2022 215 $a1 online resource (220 pages) 225 1 $aLecture notes in mathematics ;$vVolume 2305 311 08$aPrint version: Nicola, Fabio Wave Packet Analysis of Feynman Path Integrals Cham : Springer International Publishing AG,c2022 9783031061851 320 $aIncludes bibliographical references and index. 327 $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 -- 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. 327 $a4.3.2 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=(2?)-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. 327 $a8.4 Analysis of the Parametrices and Convergence Results -- 8.5 Higher-Order Parametrices -- 9 Convergence in the Lp Setting -- 9.1 Summary -- 9.2 Review of the Short Time Analysis in the Smooth Category -- 9.3 Wave Packet Analysis of the Schrödinger Flow -- 9.4 Convergence in Lp with Loss of Derivatives -- 9.5 The Case of Magnetic Fields -- 9.6 Sharpness of the Results -- 9.7 Extensions to the Case of Rough Potentials -- Bibliography -- Index. 410 0$aLecture notes in mathematics (Springer-Verlag) ;$vVolume 2305. 606 $aFeynman integrals 606 $aGabor transforms 606 $aQuantum theory 606 $aIntegrals de Feynman$2thub 606 $aTeoria quàntica$2thub 608 $aLlibres electrònics$2thub 615 0$aFeynman integrals. 615 0$aGabor transforms. 615 0$aQuantum theory. 615 7$aIntegrals de Feynman 615 7$aTeoria quàntica 676 $a515.43 700 $aNicola$b Fabio$01252174 702 $aTrapasso$b S. Ivan 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996483172503316 996 $aWave packet analysis of Feynman path integrals$92999685 997 $aUNISA