• Mathematics

Semiclassical Analysis


Author: Maciej Zworski
Publisher: American Mathematical Soc.
ISBN: 0821883208
Category: Mathematics
Page: 431
View: 1828
This book is an excellent, comprehensive introduction to semiclassical analysis. I believe it will become a standard reference for the subject. --Alejandro Uribe, University of Michigan Semiclassical analysis provides PDE techniques based on the classical-quantum (particle-wave) correspondence. These techniques include such well-known tools as geometric optics and the Wentzel-Kramers-Brillouin approximation. Examples of problems studied in this subject are high energy eigenvalue asymptotics and effective dynamics for solutions of evolution equations. From the mathematical point of view, semiclassical analysis is a branch of microlocal analysis which, broadly speaking, applies harmonic analysis and symplectic geometry to the study of linear and nonlinear PDE. The book is intended to be a graduate level text introducing readers to semiclassical and microlocal methods in PDE. It is augmented in later chapters with many specialized advanced topics which provide a link to current research literature.

    • Fourier integral operators

Semi-classical Analysis


Author: Victor Guillemin,Shlomo Sternberg
Publisher: N.A
ISBN: 9781571462763
Category: Fourier integral operators
Page: 446
View: 5840

    • Mathematics

Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics


Author: Bernard Helffer
Publisher: World Scientific
ISBN: 9789812380982
Category: Mathematics
Page: 179
View: 7218
This important book explains how the technique of Witten Laplacians may be useful in statistical mechanics. It considers the problem of analyzing the decay of correlations, after presenting its origin in statistical mechanics. In addition, it compares the Witten Laplacian approach with other techniques, such as the transfer matrix approach and its semiclassical analysis. The author concludes by providing a complete proof of the uniform Log-Sobolev inequality.

    • Science

Semi-Classical Analysis for the Schrödinger Operator and Applications


Author: Bernard Helffer
Publisher: Springer
ISBN: 3540459138
Category: Science
Page: 110
View: 5113
This introduction to semi-classical analysis is an extension of a course given by the author at the University of Nankai. It presents for some of the standard cases presented in quantum mechanics books a rigorous study of the tunneling effect, as an introduction to recent research work. The book may be read by a graduate student familiar with the classic book of Reed-Simon, and for some chapters basic notions in differential geometry. The mathematician will find here a nice application of PDE techniques and the physicist will discover the precise link between approximate solutions (B.K.W. constructions) and exact eigenfunctions (in every dimension). An application to Witten's approach for the proof of the Morse inequalities is given, as are recent results for the Schrödinger operator with periodic potentials.

    • Mathematics

Semiclassical Analysis for Diffusions and Stochastic Processes


Author: Vasily Kolokoltsov
Publisher: Springer
ISBN: 3540465871
Category: Mathematics
Page: 356
View: 3189
The monograph is devoted mainly to the analytical study of the differential, pseudo-differential and stochastic evolution equations describing the transition probabilities of various Markov processes. These include (i) diffusions (in particular,degenerate diffusions), (ii) more general jump-diffusions, especially stable jump-diffusions driven by stable Lévy processes, (iii) complex stochastic Schrödinger equations which correspond to models of quantum open systems. The main results of the book concern the existence, two-sided estimates, path integral representation, and small time and semiclassical asymptotics for the Green functions (or fundamental solutions) of these equations, which represent the transition probability densities of the corresponding random process. The boundary value problem for Hamiltonian systems and some spectral asymptotics ar also discussed. Readers should have an elementary knowledge of probability, complex and functional analysis, and calculus.

    • Mathematics

An Introduction to Semiclassical and Microlocal Analysis


Author: André Bach
Publisher: Springer Science & Business Media
ISBN: 1475744951
Category: Mathematics
Page: 191
View: 647
This book presents the techniques used in the microlocal treatment of semiclassical problems coming from quantum physics in a pedagogical, way and is mainly addressed to non-specialists in the subject. It is based on lectures taught by the author over several years, and includes many exercises providing outlines of useful applications of the semi-classical theory.

    • Mathematics

Spectral Asymptotics in the Semi-Classical Limit


Author: M. Dimassi,J. Sjostrand
Publisher: Cambridge University Press
ISBN: 9780521665445
Category: Mathematics
Page: 227
View: 754
This book presents the basic methods and applications in semiclassical approximation in the light of developments.

    • Mathematics

Semi-classical Analysis for Nonlinear Schr”dinger Equations


Author: R‚mi Carles
Publisher: World Scientific
ISBN: 9812793127
Category: Mathematics
Page: 243
View: 4839
These lecture notes review recent results on the high-frequency analysis of nonlinear Schr”dinger equations in the presence of an external potential. The book consists of two relatively independent parts: WKB analysis, and caustic crossing. In the first part, the basic linear WKB theory is constructed and then extended to the nonlinear framework. The most difficult supercritical case is discussed in detail, together with some of its consequences concerning instability phenomena. Applications of WKB analysis to functional analysis, in particular to the Cauchy problem for nonlinear Schr”dinger equations, are also given. In the second part, caustic crossing is described, especially when the caustic is reduced to a point, and the link with nonlinear scattering operators is investigated.These notes are self-contained and combine selected articles written by the author over the past ten years in a coherent manner, with some simplified proofs. Examples and figures are provided to support the intuition, and comparisons with other equations such as the nonlinear wave equation are provided.


    • Mathematics

Microlocal Analysis for Differential Operators

An Introduction
Author: Alain Grigis,Johannes Sjöstrand
Publisher: Cambridge University Press
ISBN: 9780521449861
Category: Mathematics
Page: 151
View: 1522
This book corresponds to a graduate course given many times by the authors, and should prove to be useful to mathematicians and theoretical physicists.

    • Mathematics

An Introduction to Semiclassical and Microlocal Analysis


Author: André Bach
Publisher: Springer Science & Business Media
ISBN: 1475744951
Category: Mathematics
Page: 191
View: 2392
This book presents the techniques used in the microlocal treatment of semiclassical problems coming from quantum physics in a pedagogical, way and is mainly addressed to non-specialists in the subject. It is based on lectures taught by the author over several years, and includes many exercises providing outlines of useful applications of the semi-classical theory.

    • Mathematics

Fourier Integral Operators


Author: J.J. Duistermaat
Publisher: Springer Science & Business Media
ISBN: 9780817681081
Category: Mathematics
Page: 142
View: 1231
This volume is a useful introduction to the subject of Fourier Integral Operators and is based on the author’s classic set of notes. Covering a range of topics from Hörmander’s exposition of the theory, Duistermaat approaches the subject from symplectic geometry and includes application to hyperbolic equations (= equations of wave type) and oscillatory asymptotic solutions which may have caustics. This text is suitable for mathematicians and (theoretical) physicists with an interest in (linear) partial differential equations, especially in wave propagation, rep. WKB-methods.

    • Mathematics

Microlocal Analysis and Precise Spectral Asymptotics


Author: Victor Ivrii
Publisher: Springer Science & Business Media
ISBN: 9783540627807
Category: Mathematics
Page: 731
View: 3598
Devoted to the methods of microlocal analysis applied to spectral asymptotics with accurate remainder estimates, this long awaited book develops the very powerful machinery of local and microlocal semiclassical spectral asymptotics, as well as methods of combining these asymptotics with spectral estimates. The rescaling technique, an easy to use and very powerful tool, is presented. Many theorems, considered till now as independent and difficult, are now just special cases of easy corollaries of the theorems proved in this book. Most of the results and their proofs are as yet unpublished. Part 1 considers semiclassical microlocal analysis and propagation of singularities inside the domain and near the boundary. Part 2 is on local and microlocal semiclassical spectral asymptotics for general operators and Schrödinger and Dirac operators. After a synthesis in Part 3, the real fun begins in Part 4: the main theorems are applied and numerous results, both known and new, are recovered with little effort. Then, in Chapter 12, non-Weyl asymptotics are obtained for operators in domains with thick cusps, degenerate operators, for spectral Riesz means for operators with singularities. Most of the results and almost all the proofs were never published.

    • Mathematics

Singularities of integrals

Homology, hyperfunctions and microlocal analysis
Author: Frédéric Pham
Publisher: Springer Science & Business Media
ISBN: 9780857296030
Category: Mathematics
Page: 217
View: 7232
Bringing together two fundamental texts from Frédéric Pham’s research on singular integrals, the first part of this book focuses on topological and geometrical aspects while the second explains the analytic approach. Using notions developed by J. Leray in the calculus of residues in several variables and R. Thom’s isotopy theorems, Frédéric Pham’s foundational study of the singularities of integrals lies at the interface between analysis and algebraic geometry, culminating in the Picard-Lefschetz formulae. These mathematical structures, enriched by the work of Nilsson, are then approached using methods from the theory of differential equations and generalized from the point of view of hyperfunction theory and microlocal analysis. Providing a ‘must-have’ introduction to the singularities of integrals, a number of supplementary references also offer a convenient guide to the subjects covered. This book will appeal to both mathematicians and physicists with an interest in the area of singularities of integrals. Frédéric Pham, now retired, was Professor at the University of Nice. He has published several educational and research texts. His recent work concerns semi-classical analysis and resurgent functions.

    • Technology & Engineering

Computational Electronics

Semiclassical and Quantum Device Modeling and Simulation
Author: Dragica Vasileska,Stephen M. Goodnick,Gerhard Klimeck
Publisher: CRC Press
ISBN: 1420064843
Category: Technology & Engineering
Page: 782
View: 9851
Starting with the simplest semiclassical approaches and ending with the description of complex fully quantum-mechanical methods for quantum transport analysis of state-of-the-art devices, Computational Electronics: Semiclassical and Quantum Device Modeling and Simulation provides a comprehensive overview of the essential techniques and methods for effectively analyzing transport in semiconductor devices. With the transistor reaching its limits and new device designs and paradigms of operation being explored, this timely resource delivers the simulation methods needed to properly model state-of-the-art nanoscale devices. The first part examines semiclassical transport methods, including drift-diffusion, hydrodynamic, and Monte Carlo methods for solving the Boltzmann transport equation. Details regarding numerical implementation and sample codes are provided as templates for sophisticated simulation software. The second part introduces the density gradient method, quantum hydrodynamics, and the concept of effective potentials used to account for quantum-mechanical space quantization effects in particle-based simulators. Highlighting the need for quantum transport approaches, it describes various quantum effects that appear in current and future devices being mass-produced or fabricated as a proof of concept. In this context, it introduces the concept of effective potential used to approximately include quantum-mechanical space-quantization effects within the semiclassical particle-based device simulation scheme. Addressing the practical aspects of computational electronics, this authoritative resource concludes by addressing some of the open questions related to quantum transport not covered in most books. Complete with self-study problems and numerous examples throughout, this book supplies readers with the practical understanding required to create their own simulators.

    • Mathematics

Analysis


Author: Elliott H. Lieb,Michael Loss
Publisher: American Mathematical Soc.
ISBN: 0821827839
Category: Mathematics
Page: 346
View: 4521
This is an excellent textbook on analysis and it has several unique features: Proofs of heat kernel estimates, the Nash inequality and the logarithmic Sobolev inequality are topics that are seldom treated on the level of a textbook. Best constants in several inequalities, such as Young's inequality and the logarithmic Sobolev inequality, are also included. A thorough treatment of rearrangement inequalities and competing symmetries appears in book form for the first time. There is an extensive treatment of potential theory and its applications to quantum mechanics, which, again, is unique at this level. Uniform convexity of $L^p$ space is treated very carefully. The presentation of this important subject is highly unusual for a textbook. All the proofs provide deep insights into the theorems. This book sets a new standard for a graduate textbook in analysis. --Shing-Tung Yau, Harvard University For some number of years, Rudin's ``Real and Complex'', and a few other analysis books, served as the canonical choice for the book to use, and to teach from, in a first year grad analysis course. Lieb-Loss offers a refreshing alternative: It begins with a down-to-earth intro to measure theory, $L^p$ and all that ... It aims at a wide range of essential applications, such as the Fourier transform, and series, inequalities, distributions, and Sobolev spaces--PDE, potential theory, calculus of variations, and math physics (Schrodinger's equation, the hydrogen atom, Thomas-Fermi theory ... to mention a few). The book should work equally well in a one-, or in a two-semester course. The first half of the book covers the basics, and the rest will be great for students to have, regardless of whether or not it gets to be included in a course. --Palle E. T. Jorgensen, University of Iowa

    • Science

Quantum Tunneling in Complex Systems

The Semiclassical Approach
Author: Joachim Ankerhold
Publisher: Springer
ISBN: 3540680764
Category: Science
Page: 210
View: 3126
In the last two decades remarkable progress has been made in understanding and describing tunneling processes in complex systems in terms of classical trajectories. This book introduces recent concepts and achievements. There is particular emphasis on a dynamical formulation and relations to specific systems in mesoscopic, molecular, atomic and nuclear physics.

    • Mathematics

First Summer School in Analysis and Mathematical Physics

Quantization, the Segal-Bargmann Transform, and Semiclassical Analysis : First Summer School in Analysis and Mathematical Physics, Cuernavaca Morelos, Mexico, June 8-18, 1998
Author: Salvador Pérez-Esteva,Carlos Villegas-Blas
Publisher: American Mathematical Soc.
ISBN: 0821821156
Category: Mathematics
Page: 132
View: 7713
The first Summer School of Analysis and Mathematical Physics of the Universidad Nacional Autonoma de Mexico (Cuernavaca) offered graduate and advanced undergraduate students courses on modern topics in the overlap between analysis and physics. This volume contains the expanded notes from the lectures by Brian Hall, Alejandro Uribe, and David Borthwick. The articles introduce readers to mathematical methods of classical and quantum mechanics and the link between these two theories: quantization and semiclassical analysis. Hall writes about holomorphic methods in analysis and mathematical physics and includes exercises. Uribe's lectures covered trace formulae, in particular asymptotic behavior and the relationship between the asymptotics and the geometric properties of the classical system. Borthwick presents an introduction to Kahler quantization, including the moment map, the orbit method, and symmetry and reduction. The exposition in the entire volume is geared to introducing graduate students with a basic knowledge of mathematics into areas of active research.