Heat Kernels for Elliptic and Sub-Elliptic Operators: Methods and Techniques

ISBN-13: 9780817649944 / Angielski / Twarda / 2010 / 436 str.

Ovidiu Calin; Der-Chen Chang; Kenro Furutani

Opis

With each methodology treated in its own chapter, this monograph is a thorough exploration of several theories that can be used to find explicit formulas for heat kernels for both elliptic and sub-elliptic operators. The authors show how to find heat kernels for classical operators by employing a number of different methods. Some of these methods come from stochastic processes, others from quantum physics, and yet others are purely mathematical. Depending on the symmetry, geometry and ellipticity, some methods are more suited for certain operators rather than others. What is new about this work is the sheer diversity of methods that are used to compute the heat kernels. It is interesting that such apparently distinct branches of mathematics, including stochastic processes, differential geometry, special functions, quantum mechanics and PDEs, all have a common concept - the heat kernel. This unifying concept, that brings together so many domains of mathematics, deserves dedicated study. One of the large classes of operators studied in this book is the sum of squares operators. These operators might be either elliptic or sub-elliptic. The methods for finding the heat kernel depend on the commutativity condition of the operators. Another class of operators investigated in this book is the sum between a second partially differential operator and a smooth potential. The authors demonstrate that the case of linear and quadratic potentials can be solved explicitly either by path integral methods, or by Van Vleck's formula, or by geometric methods that encounter classical action and volume function. They can also be solved by means of psuedo-differential operators. The book contains most of the heat kernels computable by means of elementary functions. Future research in this field can consider the possibility of closed-form expressions of heat kernels involving elliptic functions and hyperelliptic functions. Heat Kernels for Elliptic and Sub-elliptic Operators is an ideal resource for graduate students, researchers, and practitioners in pure and applied mathematics as well as theoretical physicists interested in understanding different ways of approaching evolution operators.

Recenzje

From the reviews:

"The present book provides a comprehensive presentation of several theories for finding explicit formulas for heat kernels for both elliptic and sub-elliptic operators. One of the nice features of the book is the diversity of methods used, coming from the theory of stochastic processes, differential geometry, special functions, quantum mechanics and PDEs. ... the book is very well organized and essentially self-contained. Hence, it is perfect reference material for graduate students and researchers in harmonic analysis and sub-Riemannian geometry, as well as theoretical physicists." (Fabio Nicola, Mathematical Reviews, Issue 2011 i)

"Authors of the present monograph devote themselves to finding the explicit formulas of heat kernels for elliptic and sub-elliptic operators. ... there are plenty of exact formulas of the heat kernels for elliptic and sub-elliptic operators in this work. Most of them are represented by means of elementary functions. ... Those results in this book are important to the experts for further studying the diffusion phenomena or the properties of solutions of parabolic equations with initial data. ... this is also a good reference book." (Jun-Qi Hu, Zentralblatt MATH, Vol. 1207, 2011)

Spis treści

Part I. Traditional Methods for Computing Heat Kernels.- Introduction.- Stochastic Analysis Method.- A Brief Introduction to Calculus of Variations.- The Path Integral Approach.- The Geometric Method.- Commuting Operators.- Fourier Transform Method.- The Eigenfunctions Expansion Method.- Part II. Heat Kernel on Nilpotent Lie Groups and Nilmanifolds.- Laplacians and Sub-Laplacians.- Heat Kernels for Laplacians and Step 2 Sub-Laplacians.- Heat Kernel for Sub-Laplacian on the Sphere S^3.- Part III. Laguerre Calculus and Fourier Method.- Finding Heat Kernels by Using Laguerre Calculus.- Constructing Heat Kernel for Degenerate Elliptic Operators.- Heat Kernel for the Kohn Laplacian on the Heisenberg Group.- Part IV. Pseudo-Differential Operators.- The Psuedo-Differential Operators Technique.- Bibliography.- Index.

Blurb

This monograph is a unified presentation of several theories of finding explicit formulas for heat kernels for both elliptic and sub-elliptic operators. These kernels are important in the theory of parabolic operators because they describe the distribution of heat on a given manifold as well as evolution phenomena and diffusion processes.

The work is divided into four main parts: Part I treats the heat kernel by traditional methods, such as the Fourier transform method, paths integrals, variational calculus, and eigenvalue expansion; Part II deals with the heat kernel on nilpotent Lie groups and nilmanifolds; Part III examines Laguerre calculus applications; Part IV uses the method of pseudo-differential operators to describe heat kernels.

Topics and features:

•comprehensive treatment from the point of view of distinct branches of mathematics, such as stochastic processes, differential geometry, special functions, quantum mechanics, and PDEs;

•novelty of the work is in the diverse methods used to compute heat kernels for elliptic and sub-elliptic operators;

•most of the heat kernels computable by means of elementary functions are covered in the work;

•self-contained material on stochastic processes and variational methods is included.

Heat Kernels for Elliptic and Sub-elliptic Operators is an ideal reference for graduate students, researchers in pure and applied mathematics, and theoretical physicists interested in understanding different ways of approaching evolution operators.