Mathematical Surveys and Monographs, Volume: 153
2009; 202 pp; hardcover
ISBN-13: 978-0-8218-4784-8
Expected publication date is March 11, 2009.
Since the early part of the twentieth century, the use of integral equations has developed into a range of tools for the study of partial differential equations. This includes the use of single- and double-layer potentials to treat classical boundary value problems.
The aim of this book is to give a self-contained presentation of an asymptotic theory for eigenvalue problems using layer potential techniques with applications in the fields of inverse problems, band gap structures, and optimal design, in particular the optimal design of photonic and phononic crystals. Throughout this book, it is shown how powerful the layer potentials techniques are for solving not only boundary value problems but also eigenvalue problems if they are combined with the elegant theory of Gohberg and Sigal on meromorphic operator-valued functions. The general approach in this book is developed in detail for eigenvalue problems for the Laplacian and the Lame system in the following two situations: one under variation of domains or boundary conditions and the other due to the presence of inclusions.
The book will be of interest to researchers and graduate students working in the fields of partial differential equations, integral equations, and inverse problems. Researchers in engineering and physics may also find this book helpful.
Graduate students and research mathematicians interested in PDE's, integral equations, and spectral analysis.
IAS/Park City Mathematics Series, Volume: 14
2009; approx. 408 pp; hardcover
ISBN-13: 978-0-8218-4765-7
Expected publication date is May 8, 2009.
Each summer the IAS/Park City Mathematics Institute Graduate Summer School gathers some of the best researchers and educators in a particular field to present lectures on a major area of mathematics. A unifying theme of the mathematical biology courses presented here is that the study of biology involves dynamical systems. Introductory chapters by Jim Keener and Mark Lewis describe the biological dynamics of reactions and of spatial processes.
Each remaining chapter stands alone, as a snapshot of in-depth research within a sub-area of mathematical biology. Jim Cushing writes about the role of nonlinear dynamical systems in understanding complex dynamics of insect populations. Epidemiology, and the interplay of data and differential equations, is the subject of David Earn's chapter on dynamic diseases. Topological methods for understanding dynamical systems are the focus of the chapter by Leon Glass on perturbed biological oscillators. Helen Byrne introduces the reader to cancer modeling and shows how mathematics can describe and predict complex movement patterns of tumors and cells. In the final chapter, Paul Bressloff couples nonlinear dynamics to nonlocal oscillations, to provide insight to the form and function of the brain.
The book provides a state-of-the-art picture of some current research in mathematical biology. Our hope is that the excitement and richness of the topics covered here will encourage readers to explore further in mathematical biology, pursuing these topics and others on their own.
The level is appropriate for graduate students and research scientists. Each chapter is based on a series of lectures given by a leading researcher and develops methods and theory of mathematical biology from first principles. Exercises are included for those who wish to delve further into the material.
Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.
Graduate students and research mathematicians interested in mathematical biology.
M. A. Lewis and J. Keener -- Introduction
J. P. Keener -- Introduction to dynamics of biological systems
M. A. Lewis, T. Hillen, and F. Lutscher -- Spatial dynamics in ecology
J. M. Cushing -- Matrix models and population dynamics
D. J. D. Earn -- Mathematical epidemiology of infectious diseases
L. Glass -- Topological approaches to biological dynamics
H. Byrne -- Mathematical modelling of solid tumour growth: from avascular to vascular, via angiogenesis
P. C. Bressloff -- Lectures in mathematical neuroscience
Student Mathematical Library, Volume: 47
2009; approx. 242 pp; softcover
ISBN-13: 978-0-8218-4699-5
Expected publication date is April 2, 2009.
This book is based on notes from the course developed and taught for more than 30 years at the Department of Mathematics of Leningrad University. The goal of the course was to present the basics of quantum mechanics and its mathematical content to students in mathematics. This book differs from the majority of other textbooks on the subject in that much more attention is paid to general principles of quantum mechanics. In particular, the authors describe in detail the relation between classical and quantum mechanics. When selecting particular topics, the authors emphasize those that are related to interesting mathematical theories. In particular, the book contains a discussion of problems related to group representation theory and to scattering theory.
This book is rather elementary and concise, and it does not require prerequisites beyond the standard undergraduate mathematical curriculum. It is aimed at giving a mathematically oriented student the opportunity to grasp the main points of quantum theory in a mathematical framework.
Undergraduate and graduate students interested in learning the basics of quantum mechanics.
The algebra of observables in classical mechanics
States
Liouville's theorem, and two pictures of motion in classical mechanics
Physical bases of quantum mechanics
A finite-dimensional model of quantum mechanics
States in quantum mechanics
Heisenberg uncertainty relations
Physical meaning of the eigenvalues and eigenvectors of observables
Two pictures of motion in quantum mechanics. The Schrodinger equation. Stationary states
Quantum mechanics of real systems. The Heisenberg commutation relations
Coordinate and momentum representations
"Eigenfunctions" of the operators Q and P
The energy, the angular momentum, and other examples of observables
The interconnection between quantum and classical mechanics. Passage to the limit from quantum mechanics to classical mechanics
One-dimensional problems of quantum mechanics. A free one-dimensional particle
The harmonic oscillator
The problem of the oscillator in the coordinate representation
Representation of the states of a one-dimensional particle in the sequence space l_2
Representation of the states for a one-dimensional particle in the space \mathcal{D} of entire analytic functions
The general case of one-dimensional motion
Three-dimensional problems in quantum mechanics. A three-dimensional free particle
A three-dimensional particle in a potential field
Angular momentum
The rotation group
Representations of the rotation group
Spherically symmetric operators
Representation of rotations by 2\times2 unitary matrices
Representation of the rotation group on a space of entire analytic functions of two complex variables
Uniqueness of the representations D_j
Representations of the rotation group on the space L^2(S^2). Spherical functions
The radial Schrodinger equation
The hydrogen atom. The alkali metal atoms
Perturbation theory
The variational principle
Scattering theory. Physical formulation of the problem
Scattering of a one-dimensional particle by a potential barrier
Physical meaning of the solutions \psi_1 and \psi_2
Scattering by a rectangular barrier
Scattering by a potential center
Motion of wave packets in a central force field
The integral equation of scattering theory
Derivation of a formula for the cross-section
Abstract scattering theory
Properties of commuting operators
Representation of the state space with respect to a complete set of observables
Spin
Spin of a system of two electrons
Systems of many particles. The identity principle
Symmetry of the coordinate wave functions of a system of two electrons. The helium atom
Multi-electron atoms. One-electron approximation
The self-consistent field equations
Mendeleev's periodic system of the elements
Lagrangian formulation of classical mechanics
Graduate Studies in Mathematics, Volume: 99
2009; approx. 302 pp; hardcover
ISBN-13: 978-0-8218-4660-5
Expected publication date is April 18, 2009.
Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators.
Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory.
This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics.
Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. It is well suited for self-study and includes numerous exercises (many with hints).
Graduate students and research mathematicians interested in mathematical physics and quantum mechanics.
Preliminaries
A first look at Banach and Hilbert spaces
Mathematical foundations of quantum mechanics
Hilbert spaces
Self-adjointness and spectrum
The spectral theorem
Applications of the spectral theorem
Quantum dynamics
Perturbation theory for self-adjoint operators
Schrodinger operators
The free Schrodinger operator
Algebraic methods
One dimensional Schrodinger operators
One-particle Schrodinger operators
Atomic Schrodinger operators
Scattering theory
Appendix
Almost everything about Lebesgue integration
Bibliographical notes
Bibliography
Glossary of notation
Index
Contemporary Mathematics, Volume: 481
2009; 196 pp; softcover
ISBN-13: 978-0-8218-4460-1
Expected publication date is April 11, 2009.
In recent years, the interplay between the methods of functional analysis and complex analysis has led to some remarkable results in a wide variety of topics. It turned out that the structure of spaces of holomorphic functions is fundamentally linked to certain invariants initially defined on abstract Frechet spaces as well as to the developments in pluripotential theory.
The aim of this volume is to document some of the original contributions to this topic presented at a conference held at Sabanc? University in ?stanbul, in September 2007. This volume also contains some surveys that give an overview of the state of the art and initiate further research in the interplay between functional and complex analysis.
Graduate students and research mathematicians interested in functional analysis, complex analysis, and linear partial differential equations.
Contemporary Mathematics, Volume: 482
2009; 240 pp; softcover
ISBN-13: 978-0-8218-4627-8
Expected publication date is April 12, 2009.
This volume represents the talks given at the Conference on Interactions between Representation Theory, Quantum Field Theory, Category Theory, Mathematical Physics, and Quantum Information Theory, held in September 2007 at the University of Texas at Tyler.
The papers in this volume, written by top experts in the field, address physical aspects, mathematical aspects, and foundational issues of quantum computation.
This volume will benefit researchers interested in advances in quantum computation and communication, as well as graduate students who wish to enter the field of quantum computation.
Graduate students and research mathematicians interested in quantum computations and quantum information theory.