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

Heat Kernels for Elliptic and Sub-elliptic Operators: Methods and Techniques
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Field name Details
Dewey Class 515.353 (DDC 23)
Title Heat Kernels for Elliptic and Sub-elliptic Operators (EB) : Methods and Techniques / by Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki.
Author Calin, Ovidiu
Added Personal Name Chang, Der-Chen author.
Furutani, Kenro author.
Iwasaki, Chisato author.
Other name(s) SpringerLink (Online service)
Edition statement edition
Publication Boston : : Birkhäuser, Boston, , 2011.
Physical Details XVIII, 436p. 25 illus. : online resource.
Series Applied and numerical harmonic analysis
ISBN 9780817649951
Summary Note 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: â¢comprehensive treatment from the point of view of distinct branches of mathematics, such as stochastic processes, differential geometry, special functions, quantum mechanics, and PDEs; â¢novelty of the work is in the diverse methods used to compute heat kernels for elliptic and sub-elliptic operators; â¢most of the heat kernels computable by means of elementary functions are covered in the work; â¢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.:
Contents note 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.
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