Semiclassical Analysis

Author: Maciej Zworski
Publisher: American Mathematical Soc.
ISBN: 9780821883204
Release Date: 2012
Genre: Mathematics

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.

Statistische Physik und Thermodynamik

Author: Walter Grimus
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 9783110414677
Release Date: 2015-09-25
Genre: Science

The updated and enlarged 2nd edition of this textbook is equally suitable for use as an enhancement to a course on statistical physics and thermodynamics or for self-study. The work focuses on presenting the terms and concepts in this broad subject area as well as on describing the systems of non-interacting particles in thermal equilibrium.

Geometric Methods in Physics

Author: Piotr Kielanowski
Publisher: Springer
ISBN: 9783319062488
Release Date: 2014-08-19
Genre: Mathematics

The Białowieża Workshops on Geometric Methods in Physics, which are hosted in the unique setting of the Białowieża natural forest in Poland, are among the most important meetings in the field. Every year some 80 to 100 participants from both the mathematics and physics world join to discuss new developments and to exchange ideas. The current volume was produced on the occasion of the 32nd meeting in 2013. It is now becoming a tradition that the Workshop is followed by a School on Geometry and Physics, which consists of advanced lectures for graduate students and young researchers. Selected speakers at the 2013 Workshop were asked to contribute to this book, and their work was supplemented by additional review articles. The selection shows that, despite its now long tradition, the workshop remains at the cutting edge of research. The 2013 Workshop also celebrated the 75th birthday of Daniel Sternheimer, and on this occasion the discussion mainly focused on his contributions to mathematical physics such as deformation quantization, Poisson geometry, symplectic geometry and non-commutative differential geometry.

Function Spaces and Partial Differential Equations

Author: Ali Taheri
Publisher: Oxford University Press, USA
ISBN: 9780198733157
Release Date: 2015
Genre: Differential equations, Partial

This is a book written primarily for graduate students and early researchers in the fields of Analysis and Partial Differential Equations (PDEs). Coverage of the material is essentially self-contained, extensive and novel with great attention to details and rigour. The strength of the book primarily lies in its clear and detailed explanations, scope and coverage, highlighting and presenting deep and profound inter-connections between different related and seemingly unrelated disciplines within classical and modern mathematics and above all the extensive collection of examples, worked-out and hinted exercises. There are well over 700 exercises of varying level leading the reader from the basics to the most advanced levels and frontiers of research. The book can be used either for independent study or for a year-long graduate level course. In fact it has its origin in a year-long graduate course taught by the author in Oxford in 2004-5 and various parts of it in other institutions later on. A good number of distinguished researchers and faculty in mathematics worldwide have started their research career from the course that formed the basis for this book.

Eigenfunctions of the Laplacian on a Riemannian Manifold

Author: Steve Zelditch
Publisher: American Mathematical Soc.
ISBN: 9781470410377
Release Date: 2017-12-12
Genre: Eigenfunctions

Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The local analysis of eigenfunctions pertains to the behavior of the eigenfunctions on wavelength scale balls. After re-scaling to a unit ball, the eigenfunctions resemble almost-harmonic functions. Global analysis refers to the use of wave equation methods to relate properties of eigenfunctions to properties of the geodesic flow. The emphasis is on the global methods and the use of Fourier integral operator methods to analyze norms and nodal sets of eigenfunctions. A somewhat unusual topic is the analytic continuation of eigenfunctions to Grauert tubes in the real analytic case, and the study of nodal sets in the complex domain. The book, which grew out of lectures given by the author at a CBMS conference in 2011, provides complete proofs of some model results, but more often it gives informal and intuitive explanations of proofs of fairly recent results. It conveys inter-related themes and results and offers an up-to-date comprehensive treatment of this important active area of research.

First Summer School in Analysis and Mathematical Physics

Author: Salvador Pérez-Esteva
Publisher: American Mathematical Soc.
ISBN: 9780821821152
Release Date: 2000
Genre: Mathematics

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.

Analysis

Author: Elliott H. Lieb
Publisher: American Mathematical Soc.
ISBN: 9780821827833
Release Date: 2001
Genre: Mathematics

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

Microlocal Analysis and Spectral Theory

Author: Luigi Rodino
Publisher: Springer Science & Business Media
ISBN: 0792345444
Release Date: 1997
Genre: Mathematics

There has been considerable recent progress in the field of microlocal analysis. In a broad sense the subject is the modern version of the classical Fourier technique for solving partial differential equations, with the localization process taking account of dual variables. The tools of pseudo-differential operators, wave-front sets and Fourier integral operators have now conferred a mature form on the theory of linear partial differential operators in the frame of Schwartz distributions or other generalized functions. At the same time, microlocal analysis has assumed an important role as an independent part of analysis, with other applications throughout mathematics and physics, one major theme being spectral theory for the Schrödinger equation in quantum mechanics. The papers collected here emphasize the topics of microlocal methods in the study of linear PDEs (analytic-Gevrey regularity of the solutions, elliptic boundary value problems, higher microlocalization), and applications to spectral theory (Schrödinger equation, asymptotic behavior of the eigenvalues, semi-classical analysis in large dimensions and statistical mechanics). Audience: Accessible to a wide audience, including graduate students in analysis and non-specialists from mathematics and physics.

Semi Classical Analysis for the Schr dinger Operator and Applications

Author: Bernard Helffer
Publisher: Springer
ISBN: 9783540459132
Release Date: 2006-11-15
Genre: Science

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.

First Summer School in Analysis and Mathematical Physics

Author: Salvador Pérez-Esteva
Publisher: American Mathematical Soc.
ISBN: 9780821821152
Release Date: 2000
Genre: Mathematics

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.

Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrodinger Equation AM 154

Author: Spyridon Kamvissis
Publisher: Princeton University Press
ISBN: 9780691114828
Release Date: 2003-09-07
Genre: Mathematics

Providing an asymptotic analysis via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrodinger equation in the semiclassical asymptotic regime, this text exploits complete integrability to establish pointwise asymptotics for this problem's solution.

Lecture Notes on Functional Analysis

Author: Alberto Bressan
Publisher: American Mathematical Soc.
ISBN: 9780821887714
Release Date: 2013
Genre: Mathematics

This textbook is addressed to graduate students in mathematics or other disciplines who wish to understand the essential concepts of functional analysis and their applications to partial differential equations. The book is intentionally concise, presenting all the fundamental concepts and results but omitting the more specialized topics. Enough of the theory of Sobolev spaces and semigroups of linear operators is included as needed to develop significant applications to elliptic, parabolic, and hyperbolic PDEs. Throughout the book, care has been taken to explain the connections between theorems in functional analysis and familiar results of finite-dimensional linear algebra. The main concepts and ideas used in the proofs are illustrated with a large number of figures. A rich collection of homework problems is included at the end of most chapters. The book is suitable as a text for a one-semester graduate course.

Spectral Geometry

Author: Alex Barnett
Publisher: American Mathematical Soc.
ISBN: 9780821853191
Release Date: 2012
Genre: Mathematics

This volume contains the proceedings of the International Conference on Spectral Geometry, held July 19-23, 2010, at Dartmouth College, Dartmouth, New Hampshire. Eigenvalue problems involving the Laplace operator on manifolds have proven to be a consistently fertile area of geometric analysis with deep connections to number theory, physics, and applied mathematics. Key questions include the measures to which eigenfunctions of the Laplacian on a Riemannian manifold condense in the limit of large eigenvalue, and the extent to which the eigenvalues and eigenfunctions of a manifold encode its geometry. In this volume, research and expository articles, including those of the plenary speakers Peter Sarnak and Victor Guillemin, address the flurry of recent progress in such areas as quantum unique ergodicity, isospectrality, semiclassical measures, the geometry of nodal lines of eigenfunctions, methods of numerical computation, and spectra of quantum graphs. This volume also contains mini-courses on spectral theory for hyperbolic surfaces, semiclassical analysis, and orbifold spectral geometry that prepared the participants, especially graduate students and young researchers, for conference lectures.

Long Time Behaviour of Classical and Quantum Systems

Author: Sandro Graffi
Publisher: World Scientific
ISBN: 981279459X
Release Date: 2001
Genre: Science

This book is centered on the two minicourses conducted by C Liverani (Rome) and J Sjoestrand (Paris) on the return to equilibrium in classical statistical mechanics and the location of quantum resonances via semiclassical analysis, respectively. The other contributions cover related topics of classical and quantum mechanics, such as scattering theory, classical and quantum statistical mechanics, dynamical localization, quantum chaos, ergodic theory and KAM techniques. Contents: Return to Equilibrium in Classical and Quantum Systems (C Liverani); Quantum Resonances and Trapped Trajectories (J SjAstrand); Return to Thermal Equilibrium in Quantum Statistical Mechanics (V Bach); Small Oscillations in Some Nonlinear PDE''s (D Bambusi & S Paleari); The Semi-Classical Van-Vleck Formula. Application to the AharonovOCoBohm Effect (J-M Bily & D Robert); Fractal Dimensions and Quantum Evolution Associated with Sparse Potential Jacobi Matrices (J-M Combes & G Mantica.); Infinite Step Billiards (M D Esposti); Semiclassical Expansion for the Thermodynamic Limit of the Ground State Energy of Kac''s Operator (B Helffer & T Ramond); Asymptotics of Scattering Poles for Two Strictly Convex Obstacles (M Ikawa); Parabolic Dynamical Systems and Inducing (S Isola); QFT for Scalar Particles in External Fields on Riemannian Manifolds (H Isozaki); Existence and Born-Oppenheimer Asymptotics of the Total Scattering Cross-Section in IonOCoAtom Collisions (T Jecko et al.); On Asymptotic Perturbation Theory for Quantum Mechanics (G Nenciu); Destruction of the Beating Effect in a Periodically Driven Double-Well (A Sacchetti); BerezinOCoToeplitz Quantization and Berezin Transform (M Schlichenmaier). Readership: Researchers, academics and graduate students in mechanics, mathematical physics and mathematics."

Equidistribution in Number Theory An Introduction

Author: Andrew Granville
Publisher: Springer Science & Business Media
ISBN: 9781402054044
Release Date: 2007-04-08
Genre: Mathematics

This set of lectures provides a structured introduction to the concept of equidistribution in number theory. This concept is of growing importance in many areas, including cryptography, zeros of L-functions, Heegner points, prime number theory, the theory of quadratic forms, and the arithmetic aspects of quantum chaos. The volume brings together leading researchers from a range of fields who reveal fascinating links between seemingly disparate areas.