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Heat Kernels for Elliptic and Sub-elliptic Operators

Methods and Techniques

  • Ovidiu Calin
  • Der-Chen Chang
  • Kenro Furutani
  • Chisato Iwasaki

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Traditional Methods for Computing Heat Kernels

    1. Front Matter
      Pages 1-1
    2. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 3-11
    3. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 13-26
    4. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 27-70
    5. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 71-74
    6. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 75-88
    7. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 89-104
    8. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 105-144
    9. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 145-197
  3. Heat Kernel on Nilpotent Lie Groups and Nilmanifolds

    1. Front Matter
      Pages 199-199
    2. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 201-223
    3. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 225-271
    4. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 273-286
  4. Laguerre Calculus and the Fourier Method

    1. Front Matter
      Pages 287-287
    2. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 289-331
    3. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 333-348
    4. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 349-358
  5. Pseudo-Differential Operators

    1. Front Matter
      Pages 359-359
    2. Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
      Pages 361-416
  6. Back Matter
    Pages 417-436

About this book

Introduction

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.

Keywords

Brownian motion Fourier transform Laguerre calculus Van Vleck formula diffusion processes elliptic and sub-elliptic operators evolution operators heat kernel nilmanifolds nilpotent Lie groups parabolic operators psuedo-differential operators quantum mechanics quartic oscillator stochastic processes sub-Riemannian manifolds

Authors and affiliations

  • Ovidiu Calin
    • 1
  • Der-Chen Chang
    • 2
  • Kenro Furutani
    • 3
  • Chisato Iwasaki
    • 4
  1. 1.Dept. MathematicsEastern Michigan UniversityYpsilantiUSA
  2. 2.Department of Mathematics and StatisticsGeorgetown UniversityWashington, DCUSA
  3. 3.Department of MathematicsScience University of TokyoNoda, ChibaJapan
  4. 4.Department of Mathematical ScienceUniversity of HyogoHimejiJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-8176-4995-1
  • Copyright Information Springer Science+Business Media, LLC 2011
  • Publisher Name Birkhäuser Boston
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-8176-4994-4
  • Online ISBN 978-0-8176-4995-1
  • Series Print ISSN 2296-5009
  • Series Online ISSN 2296-5017
  • Buy this book on publisher's site