Heat Kernels for Elliptic and Sub-elliptic Operators: Methods and Techniques (Applied and Numerical Harmonic Analysis)

Applied and Numerical Harmonic Analysis Series Editor John J. Benedetto University of Maryland Editorial Advisory Board Akram Aldroubi Vanderbilt University

Douglas Cochran Arizona State University

Ingrid Daubechies Princeton University

Hans G. Feichtinger University of Vienna

Christopher Heil Georgia Institute of Technology James McClellan Georgia Institute of Technology Michael Unser Swiss Federal Institute of Technology, Lausanne M. Victor Wickerhauser Washington University

Murat Kunt Swiss Federal Institute of Technology, Lausanne Wim Sweldens Lucent Technologies, Bell Laboratories Martin Vetterli Swiss Federal Institute of Technology, Lausanne

Ovidiu Calin Der-Chen Chang Kenro Furutani Chisato Iwasaki

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

Ovidiu Calin Department of Mathematics Eastern Michigan University Ypsilanti, MI 48197, USA [email protected]

Kenro Furutani Department of Mathematics Science University of Tokyo 2641 Yamazaki, Noda Chiba 278-8510, Japan [email protected]

Der-Chen Chang Department of Mathematics and Statistics Georgetown University Washington, DC 20057, USA [email protected]

Chisato Iwasaki Department of Mathematical Science University of Hyogo 2167 Shosha Himeji 671-2201, Japan [email protected]

ISBN 978-0-8176-4994-4 e-ISBN 978-0-8176-4995-1 DOI 10.1007/978-0-8176-4995-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010937587 Mathematics Subject Classication (2010): 22E25, 35B40, 35Q80, 35S05, 35R03, 35P10, 35K08, 35K30, 35K65, 35K05, 35F21, 35H10, 35J08, 49L99, 53B20, 53C15, 58J65, 53C17 c Springer Science+Business Media, LLC 2011  All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper www.birkhauser-science.com

ANHA Series Preface

The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the engineering, mathematical, and scientific communities with significant developments in harmonic analysis, ranging from abstract harmonic analysis to basic applications. The title of the series reflects the importance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the interleaving of theory and applications and their creative symbiotic evolution is axiomatic. Harmonic analysis is a wellspring of ideas and applicability that has flourished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental relationship between harmonic analysis and fields such as signal processing, partial differential equations (PDEs), and image processing is reflected in our state-of-theart ANHA series. Our vision of modern harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, time-frequency analysis, and fractal geometry, as well as the diverse topics that impinge on them. For example, wavelet theory can be considered an appropriate tool to deal with some basic problems in digital signal processing, speech and image processing, geophysics, pattern recognition, biomedical engineering, and turbulence. These areas implement the latest technology from sampling methods on surfaces to fast algorithms and computer vision methods. The underlying mathematics of wavelet theory depends not only on classical Fourier analysis, but also on ideas from abstract harmonic analysis, including von Neumann algebras and the affine group. This leads to a study of the Heisenberg group and its relationship to Gabor systems, and of the metaplectic group for a meaningful interaction of signal decomposition methods. The unifying influence of wavelet theory in the aforementioned topics illustrates the justification for providing a means for centralizing and disseminating information from the broader, but still focused, area of harmonic analysis. This will be a key role of ANHA. We intend to publish with the scope and interaction that such a host of issues demands. Along with our commitment to publish mathematically significant works at the frontiers of harmonic analysis, we have a comparably strong commitment to publish v

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ANHA Series Preface

major advances in the following applicable topics in which harmonic analysis plays a substantial role: Antenna theory Prediction theory Biomedical signal processing Radar applications Digital signal processing Sampling theory Fast algorithms Spectral estimation Gabor theory and applications Speech processing Image processing Time-frequency and Numerical partial differential equations time-scale analysis Wavelet theory The above point of view for the ANHA book series is inspired by the history of Fourier analysis itself, whose tentacles reach into so many fields. In the last two centuries Fourier analysis has had a major impact on the development of mathematics, on the understanding of many engineering and scientific phenomena, and on the solution of some of the most important problems in mathematics and the sciences. Historically, Fourier series were developed in the analysis of some of the classical PDEs of mathematical physics; these series were used to solve such equations. In order to understand Fourier series and the kinds of solutions they could represent, some of the most basic notions of analysis were defined, e.g., the concept of “function.” Since the coefficients of Fourier series are integrals, it is no surprise that Riemann integrals were conceived to deal with uniqueness properties of trigonometric series. Cantor’s set theory was also developed because of such uniqueness questions. A basic problem in Fourier analysis is to show how complicated phenomena, such as sound waves, can be described in terms of elementary harmonics. There are two aspects of this problem: first, to find, or even define properly, the harmonics or spectrum of a given phenomenon, e.g., the spectroscopy problem in optics; second, to determine which phenomena can be constructed from given classes of harmonics, as done, for example, by the mechanical synthesizers in tidal analysis. Fourier analysis is also the natural setting for many other problems in engineering, mathematics, and the sciences. For example, Wiener’s Tauberian theorem in Fourier analysis not only characterizes the behavior of the prime numbers, but also provides the proper notion of spectrum for phenomena such as white light; this latter process leads to the Fourier analysis associated with correlation functions in filtering and prediction problems, and these problems, in turn, deal naturally with Hardy spaces in the theory of complex variables. Nowadays, some of the theory of PDEs has given way to the study of Fourier integral operators. Problems in antenna theory are studied in terms of unimodular trigonometric polynomials. Applications of Fourier analysis abound in signal processing, whether with the fast Fourier transform (FFT), or filter design, or the adaptive modeling inherent in time-frequency-scale methods such as wavelet theory. The coherent states of mathematical physics are translated and modulated Fourier

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transforms, and these are used, in conjunction with the uncertainty principle, for dealing with signal reconstruction in communications theory. We are back to the raison d’ˆetre of the ANHA series! University of Maryland College Park

John J. Benedetto Series Editor

Preface

The Fourier transform is known as one of the most powerful and useful methods for finding fundamental solutions of operators with constant coefficients. However, in the general case this method has its own limitations. This book presents several other methods that can be used concurrently with the Fourier transform method to obtain heat kernels for elliptic and sub-elliptic operators. The text contains a large number of examples which facilitate understanding. An Overview for the Reader. The theory of parabolic operators describes the distribution of heat on a given manifold as well as evolution phenomena and diffusion processes. The solution of an initial value problem for a parabolic partial differential equation depends on its heat kernel, which is the fundamental solution of the associated parabolic operator. Hence the importance of finding explicit formulas for these kernels. This monograph presents several theories for finding explicit formulas for heat kernels for both elliptic and sub-elliptic operators. These methods are treated in distinct chapters. We shall find heat kernels for classical operators by several different methods. Some methods come from stochastic processes, others come from quantum physics, and others are purely mathematical. Depending on the symmetry, geometry and ellipticity, some methods are more suited for certain operators rather than others. This book is a perfect reference material for graduate students, researchers in pure and applied mathematics as well as theoretical physicists interested in understanding different ways of approaching evolution operators. Scientific Outline. Heat kernels arise naturally from probabilistic properties of stochastic processes. The transition density of a stochastic process provides the heat kernel for the associated generator operator, which is a second-order PDE operator. For instance, the generator of a Brownian motion in a plane is the operator 1 2 .@ C @2y /. On the other side, since the x- and y-components of the Brownian 2 x motion are independent, the joint transition density, given that the motion starts at .x0 ; y0 / at t D 0, is the product of two transition densities: .xx0 /2 .xx0 /2 1 1 1  1 Œ.xx0 /2 C.yy0 /2  e 2t e  2t  p e  2t p D ; 2 t 2 t 2 t

which is the heat kernel for the aforementioned generator operator. ix

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Preface

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. If the operators commute, then the heat kernel is the product of the heat kernels of the operators. If the operators do not commute, then the Trotter formula applies and the heat kernel is computed using the path integral method. This method was borrowed from quantum physics, and it was used to compute propagators for the Schr¨odinger equation. This technique of path integrals was initiated by Feynman in the early 1940s. Another class of operators investigated by this book is the sum between a second partial differential operator and a smooth potential. The case of linear and quadratic potentials can be solved explicitly by the path integral method, by van Vleck’s formula, or by geometric methods that encounter classical action and volume function. Finally, they can also be solved by means of pseudo-differential operators. For instance, the aforementioned operator 12 .@2x C @2y / has a heat kernel of the type K.x0 ; y0 ; x; yI t/ D V .t/e Scl .x0 ;y0 ;x;yIt / ; where V .t/ D

1 2 t

(0.0.1)

describes the density of geodesics emerging from .x0 ; y0 /, and

Scl .x0 ; y0 ; x; yI t/ D

  1 1 Œ.x  x0 /2 C .y  y0 /2  D dist2 .x0 ; y0 /; .x; y/ 2t 2t

is the classical action on the Euclidean plane from .x0 ; y0 / to .x; y/ at time t. The key idea here is that the heat propagates mainly along the geodesics of the associated (sub-)Riemannian space. Then the heat density at a certain point in the space, which is the heat kernel of a certain operator, depends on the action along the geodesic and the density of geodesics at that point, given that all geodesics start at a given initial point. The density of geodesics is described by the volume function, which in certain special cases is given by a van Vleck determinant. This idea can be applied in general for elliptic operators to obtain locally closedform expressions for the heat kernels for certain operators. However, there are some limitations to the applicability of the method, and it depends on the nature of the potential function. The operators involving a power potential of degree greater than 2 do not in general have a closed-form solution. Geometrically speaking, this reduces to infinitely many geodesics between two points whose energies cannot be obtained in a closed-form solution in terms of the boundary points. None of the methods presented in this book can be applied to obtain exact solutions for these types of operators. For instance, the well-known problem of nonsolvability of the quartic oscillator problem is one of them. In the case of sub-elliptic operators, formula (0.0.1) no longer holds, due to the degeneracy of the operator, and another formula will take its place. This new formula will require a fiber integration along the characteristic variety (which for elliptic operators degenerate to a point). We note that pseudo-differential techniques and the path integration method do not, in general, provide easy ways of obtaining heat kernels for sub-elliptic operators.

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The novelty of this work is the diversity of methods aimed at computing heat kernels for elliptic and sub-elliptic operators. It is interesting that apparently distinct branches of mathematics, such as stochastic processes, differential geometry, special functions, quantum mechanics and PDEs, have all a common concept – the heat kernel. This concept unifies the aforementioned domains of mathematics, and hence deserves us dedicating our study to it. It is worth noting the relation of the material of this book with other previous books on the subject. One of the long-standing textbooks in the field is the well-known book of David Widder [111], which appeared in the mid-1970s. This treats the heat equation mainly in dimension 1, discussing boundary value problems, Green’s functions, integral transforms, theta-functions, the Huygens property, series expansions, heat polynomials, and other miscellaneous topics. Unlike the classical tract of the aforementioned reference, the present monograph covers the heat equation in several other contexts, such as geometric, stochastic, and quantic. However, the present work does not intend to replace Widder’s book, but to complement it with newer facts regarding the geometric and analytic contexts. This book contains most of the heat kernels computable by means of elementary functions. Future developments in this field can consider the possibility of closed-form expressions of heat kernels involving elliptic functions and hyperelliptic functions. These types of special functions have already appeared in the explicit computation of geodesics on certain sub-Riemannian manifolds, and we treated them in the monograph [27]. Similar types of functions also appeared in considering the heat kernel of operators with polynomial potential of degree greater than 2, one of the most famous being the “quartic oscillator” example. In general, the heat kernels of sub-elliptic operators associated with some sub-Riemannian manifolds of step larger than 2 may lead to the need for special functions. However, our present restrictive knowledge of the subject of hyperelliptic function theory indicates today’s limit of explicit computability of these types of heat kernels. The material was prepared as follows: Chaps. 1–8 were prepared by Ovidiu Calin; Chaps. 9–11 were written by Kenro Furutani, Chaps. 12–14 are attributed to Der-Chen Chang, while Chap. 15 was worked out by Chisato Iwasaki. Two of the authors (Calin and Chang) reside in the United States, while the others (Furutani and Iwasaki) live in Japan. Acknowledgments We wish to express our gratitude to Prof. Peter Greiner, who introduced us to the subject of this book. The monograph was written in 2009–2010 while the first author was on a sabbatical leave from Eastern Michigan University, and he was also partially supported by NSF Grant no. 0631541 and a fund from Tokyo University of Science during his stay there in the summer of 2009. The second author was partially supported by Hong Kong RGC Grant no. 600607, the Norwegian Research Council Research Grant no. 180275/D15 and a competitive research grant at Georgetown University. The third author was partially supported by the JSPS-grant and Grantsin-Aid for Scientific Research (C) no. 20540218, and the fourth author was partially supported by the Grants-in-Aid for Scientific Research (C) no. 21540194. Finally, we would like to express our thanks to the Birkh¨auser and ANHA editors, especially to J. J. Benedetto, in making this project possible.

Tokyo, 2009

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Preface The chapters’ diagram

Contents

Part I Traditional Methods for Computing Heat Kernels 1

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3 1.1 Physical Significance of the Heat Equation.. . . . . . . .. . . . . . . . . . . . . . . . . 3 1.2 The Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5 1.3 The Heat Equation in Its Setting . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 6 1.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 7 1.5 Theta-Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 7 1.6 Some Useful Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 10

2

A Brief Introduction to the Calculus of Variations . . . . . .. . . . . . . . . . . . . . . . . 2.1 Lagrangian Mechanics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.1 The First Variation . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.2 The Second Variation . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.3 Geometrical Interpretation .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.4 The Case of Riemannian Geometry . . . . . . .. . . . . . . . . . . . . . . . . 2.1.5 Examples of Lagrangian Dynamics . . . . . . .. . . . . . . . . . . . . . . . . 2.2 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.3 The Hamilton–Jacobi Equation . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

13 13 14 15 16 18 20 23 24

3

The Geometric Method . . . . . . P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1 Heat Kernel for L D 12 ni;j D1 aij @xi @xj . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.2 Interpretation of the Volume Function . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . P 3.3 The Operator L D 12 niD1 bi @2xi . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.4 The Generalized Transport Equation.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.5 Solving the Generalized Transport Equation . . . . . . .. . . . . . . . . . . . . . . . . 3.6 The Operator L D 12 y m .@2x C @2y /, m 2 N . . . . . . . . .. . . . . . . . . . . . . . . . . 3.7 Heat Kernels on Curved Spaces .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.8 Heat Kernel at the Cut-Locus . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.9 Heat Kernel at the Conjugate Locus . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.10 Heat Kernel on the Half-Line . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.11 Heat Kernel on S 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.12 Heat Kernel on the Segment Œ0; T  . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

27 27 30 31 33 38 40 45 46 47 47 48 49 xiii

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Contents

3.13 3.14 3.15 3.16 3.17 3.18

Heat Kernel on the Cylinder .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . Operators with Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . The Linear Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . The Quadratic Potential.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . P The Operator 12 @2xi ˙ 12 a2 jxj2 . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . The Operator L D 12 @2x C x2 . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

50 52 54 58 60 64

4

Commuting Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.1 Commuting Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.1.1 The Operator L D 12 .@2x C @2y / . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.1.2 The Operator L D 12 .@2x C y@2y / . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.1.3 The Operator L D 12 .x@2x C y@2y /. . . . . . . . .. . . . . . . . . . . . . . . . . 4.1.4 Sum of Squares of Linear Potentials . . . . . .. . . . . . . . . . . . . . . . . 4.1.5 The Operator L D 12 .x 2 C y 2 /.@2x C @2y / . . . . . . . . . . . . . . . . .

71 71 71 72 72 72 73

5

The Fourier Transform Method . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5.2 Heat Kernel for the Grushin Operator . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5.3 Heat Kernel for @2x C x@x @y . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5.4 The Formula of Beals, Gaveau and Greiner . . . . . . . .. . . . . . . . . . . . . . . . . 5.5 Kolmogorov Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5.6 The Operator 12 @2x C x@2y . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . P 5.7 The Generalized Grushin Operator 12 niD1 @2xi C 2jxj2 @2t . . . . . . . . . . 5.8 The Heisenberg Laplacian .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

75 75 76 77 78 80 85 85 87

6

The Eigenfunction Expansion Method .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 89 6.1 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 89 6.2 Mehler’s Formula and Applications.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 91 6.3 Hille–Hardy’s Formula and Applications .. . . . . . . . . .. . . . . . . . . . . . . . . . . 94 6.4 Poisson’s Summation Formula and Applications . .. . . . . . . . . . . . . . . . . 98 6.4.1 Heat Kernel on S 1 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 99 6.4.2 Heat Kernel on the Segment Œ0; T  . . . . . . . .. . . . . . . . . . . . . . . . .100 6.5 Legendre’s Polynomials and Applications .. . . . . . . . .. . . . . . . . . . . . . . . . .102 6.6 Legendre Functions and Applications . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .104

7

The Path Integral Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .105 7.1 Introducing Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .105 7.2 Paths Integrals via Trotter’s Formula . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .107 7.3 Formal Algorithm for Obtaining Heat Kernels .. . . .. . . . . . . . . . . . . . . . .112 7.4 The Operator 12 @2x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .118 7.5 The Operator 12 .@2x C @x / . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .123 7.6 Heat Kernel for L D 12 .@2x  @x / . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .125 7.7 Heat Kernel for L D 12 x 2 @2x C x@x . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .126 7.8 The Hermite Operator 12 .@2x  a2 x 2 / . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .126

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7.9

Evaluating Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .132 7.9.1 Van Vleck’s Formula .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .132 7.9.2 Applications of van Vleck’s Formula . . . . .. . . . . . . . . . . . . . . . .133 7.9.3 The Feynman–Kac Formula . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .138 Non-Commutativity of Sums of Squares . . . . . . . . . . .. . . . . . . . . . . . . . . . .140 Path Integrals and Sub-Elliptic Operators . . . . . . . . . .. . . . . . . . . . . . . . . . .143

7.10 7.11 8

The Stochastic Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .145 8.1 Elements of Stochastic Processes . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .145 8.2 Ito Diffusion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .152 8.3 The Generator of an Ito Diffusion.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .154 8.4 Kolmogorov’s Backward Equation and Heat Kernel .. . . . . . . . . . . . . . .156 8.5 Algorithm for Finding the Heat Kernel . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .157 8.6 Finding the Transition Density . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .159 8.7 Kolmogorov’s Forward Equation . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .166 8.8 The Operator 12 @2x  x@x . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .167 8.9 Generalized Brownian Motion . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .168 8.10 Linear Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .170 8.11 Geometric Brownian Motion .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .171 8.12 Mean Reverting Ornstein–Uhlenbeck Process . . . . .. . . . . . . . . . . . . . . . .174 8.13 Bessel Operator and Bessel Process . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .176 8.14 Brownian Motion on a Circle . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .181 8.15 An Example of a Heat Kernel . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .183 8.16 A Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .185 8.17 Kolmogorov’s Operator.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .188 8.18 The Operator 12 x12 @2x2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .189 8.19 Grushin’s Operator .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .191 8.20 Squared Bessel Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .192 8.21 CIR Processes .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .193 8.22 Limitations of the Method .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .193 8.23 Operators with Potential and the Feynman–Kac Formula . . . . . . . . . .195

Part II Heat Kernel on Nilpotent Lie Groups and Nilmanifolds 9

Laplacians and Sub-Laplacians .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .201 9.1 Sub-Riemannian Structure and Heat Kernels .. . . . . .. . . . . . . . . . . . . . . . .201 9.1.1 Sub-Riemannian Structure .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .201 9.1.2 Heat Kernel of the Sub-Laplacian and Laplacian .. . . . . . . . .203 9.1.3 The Volume Form . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .206 9.2 Laplacian and Sub-Laplacian on Nilpotent Lie Groups .. . . . . . . . . . . .207 9.2.1 Nilpotent Lie Groups.. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .207 9.2.2 The Heisenberg Group .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .209 9.2.3 Higher-Dimensional Heisenberg Groups .. . . . . . . . . . . . . . . . .211 9.2.4 Quaternionic Heisenberg Group .. . . . . . . . . .. . . . . . . . . . . . . . . . .213 9.2.5 Heisenberg-Type Lie Algebra . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .216

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9.2.6 9.2.7

Free Two-Step Nilpotent Lie Algebra .. . . .. . . . . . . . . . . . . . . . .219 The Engel Group . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .220

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians . . . . . . . . . . . . . . . .225 10.1 Spectral Decomposition and Heat Kernel.. . . . . . . . . .. . . . . . . . . . . . . . . . .225 10.2 Complex Hamilton–Jacobi Theory.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .229 10.2.1 Path Integrals and Integral Expression of a Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .230 10.2.2 A Solution of a Hamilton–Jacobi Equation . . . . . . . . . . . . . . . .232 10.2.3 The Generalized Transport Equation .. . . . .. . . . . . . . . . . . . . . . .237 10.2.4 Heat Kernel for the Sub-Laplacian . . . . . . . .. . . . . . . . . . . . . . . . .244 10.2.5 Heat Kernel for Laplacian I . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .251 10.2.6 Heat Kernel for Laplacian II . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .252 10.2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .255 10.3 Grushin-Type Operators and the Heat Kernel . . . . . .. . . . . . . . . . . . . . . . .263 10.3.1 Grushin-Type Operators . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .263 10.3.2 Heisenberg Group Case . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .265 10.3.3 Heat Kernel of the Grushin Operator .. . . . .. . . . . . . . . . . . . . . . .267 11 Heat Kernel for the Sub-Laplacian on the Sphere S 3 . . .. . . . . . . . . . . . . . . . .273 11.1 Sub-Riemannian Structure and Sub-Laplacian on the Sphere S 3 . .273 11.1.1 S 3 in the Quaternion Number Field . . . . . . .. . . . . . . . . . . . . . . . .273 11.1.2 Heat Kernel of the Sub-Laplacian on S3 . .. . . . . . . . . . . . . . . . .277 11.1.3 Spherical Grushin Operator and the Heat Kernel .. . . . . . . . .282 Part III Laguerre Calculus and the Fourier Method 12 Finding Heat Kernels Using the Laguerre Calculus . . . . .. . . . . . . . . . . . . . . . .289 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .289 12.2 Laguerre Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .291 12.2.1 Twisted Convolution . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .292 12.2.2 P.V. Convolution Operators . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .293 12.2.3 Laguerre Functions .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .294 12.2.4 Laguerre Calculus on H1 . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .295 12.2.5 Laguerre Calculus on Hn . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .296 12.2.6 Left Invariant Differential Operators . . . . . .. . . . . . . . . . . . . . . . .297 12.3 The Heisenberg Sub-Laplacian.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .300 12.4 Powers of the Sub-Laplacian .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .303 12.5 Heat Kernel for the Operator Lm ˛ . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .306 12.6 Fundamental Solution of the Paneitz Operator . . . . .. . . . . . . . . . . . . . . . .309 12.6.1 Laguerre Tensor of the Paneitz Operator ... . . . . . . . . . . . . . . . .311 12.6.2 Fundamental Solution: The Case n  2 . . .. . . . . . . . . . . . . . . . .313 12.6.3 Fundamental Solution: The Case n D 1 . .. . . . . . . . . . . . . . . . .316 12.7 Heat Kernel of the Paneitz Operator . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .318 12.8 Projection and Relative Fundamental Solution .. . . .. . . . . . . . . . . . . . . . .321

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xvii

12.9 The Kernel ‰m .z; t/ for m > n .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .326 12.10 The Isotropic Heisenberg Group . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .328 12.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .330 13 Constructing Heat Kernels for Degenerate Elliptic Operators .. . . . . . . . .333 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .333 13.2 Finding Heat Kernels for Operators Lj , j D 1; : : : ; 5 .. . . . . . . . . . . . .335 13.3 Some Explicit Calculations .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .340 13.3.1 Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .341 13.3.2 Kolmogorov Operator .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .342 13.3.3 The Operator L3 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .344 13.3.4 The Operator L4 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .346 13.3.5 The Operator L5 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .347 14 Heat Kernel for the Kohn Laplacian on the Heisenberg Group .. . . . . . . .349 14.1 The Kohn Laplacian on the Heisenberg Group .. . . .. . . . . . . . . . . . . . . . .349 14.2 Full Fourier Transform on the Group . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .351 14.3 Solving the Transformed Heat Equation Using Hermite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .353 14.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .358 Part IV

Pseudo-Differential Operators

15 The Pseudo-Differential Operator Technique . . . . . . . . . . . .. . . . . . . . . . . . . . . . .361 15.1 Basic Results of Pseudo-Differential Operators .. . .. . . . . . . . . . . . . . . . .361 15.1.1 Definition of Pseudo-Differential Operators . . . . . . . . . . . . . .362 15.1.2 Calculus with Pseudo-Differential Operators .. . . . . . . . . . . . .363 15.1.3 Proof of Lemma 15.1.3 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .366 15.2 Fundamental Solution by Symbolic Calculus: The Nondegenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .372 15.3 Basic Results for Pseudo-Differential Operators of Weyl Symbols 376 15.3.1 Calculus with Pseudo-Differential Operators .. . . . . . . . . . . . .376 15.4 Fundamental Solution by Symbolic Calculus: The Degenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .382 15.4.1 Construction of the Symbol.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .382 P 15.4.2 The Symbol pm D 12 `j D1 qj2 . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .396 15.4.3 The Special Case of Quadratic Symbols . .. . . . . . . . . . . . . . . . .397 15.4.4 A Key Theorem for Eigenfunction Expansion .. . . . . . . . . . . .400 15.5 The Hermite Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .400 15.5.1 Exact Form of the Symbol of the Fundamental Solution .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .401 15.5.2 Eigenfunction Expansion . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .403 15.6 The Grushin Operator.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .407 15.7 Exact Form of the Symbol of the Fundamental Solution for the Sub-Laplacian .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .408

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15.8 15.9

The Sub-Laplacian on Step-2 Nilpotent Lie Groups . . . . . . . . . . . . . . . .410 The Kolmogorov Operator . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .415

Appendix . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .417 Conclusions . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .419 List of Frequently Used Notations and Symbols . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .423 References .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .425 Index . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .431

Part I

Traditional Methods for Computing Heat Kernels

Chapter 1

Introduction

1.1 Physical Significance of the Heat Equation Consider the problem of finding the temperature of a gas contained in a volume V , which is supposed to be a bounded, open and connected set in R3 . Denote by u.x; t/ the gas temperature at the point x 2 V at time t. Let  be a subdomain of V with smooth boundary @. Let  be the unit normal to @; see Fig. 1.1a. In order to write the equation of temperature evolution, we shall consider the law of conservation of thermal energy on the domain , under the assumption that there is no external absorption or application of heat: The rate of change of the energy with respect to time in  is equal to the net flow of energy across the boundary @. The total thermal energy (heat) in  at time t is Z U.t/ D

u.x; t/ dx;

(1.1.1)



and the net flow of energy across the boundary is given by the Fourier law Z Q.t/ D @

@u.x; t/ d; @

where d is the area element on @. The normal derivative @u.x; t/ D hrx u.x; t/; .x/i; @

x 2 @;

is the flux density normal to the boundary @. If x 2 @, the only gas particles in a small neighborhood U of x that hit @ are those which have a normal direction to @. The other particles in U that do not have a perpendicular direction to @ do not have a contribution toward the term Q.t/, since they are colliding with other particles in U and are bouncing back: see Fig. 1.1b.

O. Calin et al., Heat Kernels for Elliptic and Sub-elliptic Operators, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-4995-1 1, c Springer Science+Business Media, LLC 2011 

3

4

1 Introduction

a

b

Fig. 1.1 .a/ The domain  and its normal . .b/ The particles which do not have normal direction do not contribute to the heat exchanged through @

Applying Gauss’s formula, the heat changed through the boundary becomes Z

Z Q.t/ D 

divrx u.x; t/ dx D



x u.x; t/ dx:

(1.1.2)

The aforementioned conservation of energy can be written as @ U.t/ D Q.t/I @t using (1.1.1) and (1.1.2), it becomes Z 

@ u.x; t/ dx D @t

Z 

x u.x; t/ dx:

Since  is an arbitrary domain of V , it follows that @ u.x; t/ D x u.x; t/; @t

x 2 V; t  0;

which is called the heat equation for the Laplacian x on R3 . A similar approach can beapplied in the case when .V; gij / is a Riemannian  manifold and ; .x1 ; : : :; xn / is a local chart on V . In this case the measure dx is p replaced by the volume element d vg D det.gij /dx1 ^  ^dxn and the divergence and the gradient are given by  X @  1 det.gij /X j ; div X D det.gij / @xi n

i D1

rx u D

X i;j

g ij

@u @ : @xj @xi

1.2 The Fundamental Solution

5

The Laplacian in this case becomes the elliptic operator LD

X

g ij

i;j

where ijk D

1 2

X `

g k`



@gi ` @xj

C

X @2 @  ijk ; @xi @xj @xk k

@gj ` @xi



@gij @x`

 are the Christoffel symbols.

1.2 The Fundamental Solution Consider the following heat problem: Given two distinct points x0 ; x on a Riemannian manifold .V; g/, assume that a unit volume of heat is applied at the point x0 at time t D 0. What is the heat distribution at any time t > 0? Equivalently, what is volume of heat K.x0 ; xI t/ that transmits from x0 to x after time t? K.x0 ; xI t/ is defined on V  V  .0; 1/ and is called the fundamental solution of the heat operator, or the heat kernel, if it satisfies the equations @ K.x0 ; xI t/ D LK.x0 ; xI t/; @t lim K.x0 ; xI t/ D ıx0 :

x 2 V; t  0;

t &0

Since in the compact case (with no boundary) no heat is lost, the total amount of heat is preserved: Z Z K.x0 ; xI t/ dx D ıx0 .x/dx D 1: This is the reason why K.x0 ; xI t/ can be considered as a probability density function for any t > 0 for a certain random process discussed in Chap. 2. For instance, in the one-dimensional case, at t D 0 the density is the Dirac distribution ıx0 centered at x0 , while at t > 0, the distribution is Gaussian; see Fig. 1.2a, b.

a y

b y

± x0 t =0 x0

x

t>0 x0

Fig. 1.2 .a/ The Dirac distribution ıx0 . .b/ The one-dimensional Gaussian

p1 e 2 t

.xx0 /2 2t

x-

6

1 Introduction

The main use of the heat kernel is to find the temperature evolution in V , provided the initial distribution of temperature is given. For any continuous, bounded function , Cauchy’s problem @ u.x; t/ D Lu.x; t/; @t u.x; 0/ D .x/;

x 2 V; t  0;

Z

has the solution u.x; t/ D

V

K.x; yI t/.y/ d vg :

Since K.x; yI t/.y/ represents the volume of heat transmitted from y to x after time t, given the initial temperature .x/, the kernel K.x; yI t/ is sometimes also called a propagator.

1.3 The Heat Equation in Its Setting The heat equation is a particular case of a more general linear second-order partial differential equation of the type A@2x C B@x @t C C @2t C D@x C E@t C F D 0; where the coefficients A; B; C; D; E; F are functions on x and t only. The following classification is familiar from undergraduate courses on differential equations: Condition B 2 < 4AC B 2 > 4AC B 2 D 4AC

Type Elliptic Hyperbolic Parabolic

Example @2x u C @2t u D 0 @2x u  @2t u D 0 @2x u  @t u D 0

Name Laplace Wave Heat

The type is suggested by the corresponding conic curve defined by the principal symbol. The Laplace equation describes static equations such as minimal surfaces, gravitational potentials, electric potentials, etc., having the Riemannian geometry as its underlined geometry. The wave operators describe the propagation of mechanical, radio and electro-magnetic waves, and radiation. It is associated with a geometry of Lorentz type. The heat equation describes diffusion phenomena in classical physics, Schr¨odinger-type equations in quantum mechanics, and the Black–Scholes equation in finance. The reader can find an elementary treatment of these equations, for instance, in Folland [44].

1.5 Theta-Functions

7

1.4 Methodology The first part of this book provides several methods for computing heat kernels for both elliptic and sub-elliptic operators using the following methods:     

Stochastic analysis of diffusion processes The Path integral method The Geometric method involving the action along the geodesics The Fourier transform method The eigenfunction expansion method

The stochastic analysis method associates with each Ito diffusion a generator operator. The heat kernel of this operator is given by a transition density for the associated Ito diffusion. Several methods for computing transition densities are provided. The path integral method stems from the computation of propagators in quantum mechanics and has the following approaches:    

By direct computation By using Trotter’s formula By using van Vleck’s formula By using Feynman–Kac’s formula

The eigenfunction expansion is a method which can be applied as long as we are able to compute the eigenfunctions and eigenvalues of the operator. However, closed-form solutions can be provided only if one of the following bilinear generating formulas is used:    

Mehler’s formula Hille–Hardy’s formula Poisson’s summation formula Any other bilinear generating function

The geometric method expresses the heat kernel in terms of the action and volume element. It is a method which can be applied successfully as long as we are able to find the explicit formula for the action and solve the transport equation. It can be applied for both elliptic and sub-elliptic operators.

1.5 Theta-Functions A theta-function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions. We shall use them in the sequel when several heat kernels will be expressed in terms of theta-functions.

8

1 Introduction

Consider the heat equation @ @2  2 D0 @t @x

(1.5.3)

on Œ0; , with constant diffusivity . There are two important boundary-type conditions which lead to several types of theta-functions. Dirichlet boundary condition. The heat is maintained at zero temperature at the endpoints at all times t: .0; t/ D .; t/ D 0: The separation-of-variables method provides the solution of (1.5.3) in the form 1 .x; t/ D 2

X

2 t

.1/n e .2nC1/

sin.2n C 1/x;

n0

also called the first theta-function. We note the temperature will decrease to zero in the long run, so limt !1 1 .x; t/ D 0. Neumann boundary condition. There is no heat exchange at the endpoints x D 0; . In this case we say the endpoints are insulated and the following condition holds at the boundary: @ @ .0; t/ D .; t/ D 0: @x @x In this case the solution of (1.5.3) is 4 .x; t/ D 1 C 2

X

.1/n e 4n

2 t

cos 2nx;

n1

called the fourth theta-function. Since the heat cannot escape the domain, in the long run the solution tends to a constant equilibrium solution. A translation of 1 and 4 with =2 in the direction of x yields two other thetafunctions: X 2 2 .x; t/ D 1 .x C =2; t/ D 2 e 4.nC1=2/ t cos.2n C 1/x; n0

3 .x; t/ D 4 .x C =2; t/ D 1 C 2

X

e 4n

2 t

cos 2nx:

n1

The theta-functions 1 and 2 are periodic with period 2, and the functions 3 and 4 are periodic with period .

1.5 Theta-Functions

9

A few variations of the definition of theta-functions can be found in the literature. If we let q D e 4t , then the aforementioned formulas become 1 .x; q/ D 2

X

2

.1/n q .nC1=2/ sin.2n C 1/x;

n0

2 .x; q/ D 2

X

2

q .nC1=2/ cos.2n C 1/x;

n0

3 .x; q/ D 1 C 2

X

2

q n cos 2nx;

n1

4 .x; q/ D 1 C 2

X

2

.1/n q n cos 2nx:

n1

The graphs of the theta-functions with q D 0:7 and x 2 Œ0; 12 can be seen in Fig. 1.3. In the aforementioned formulas the variable q can also be considered complex with jqj < 1. In this case, if we let q D e i  , with 2 C, and I m. / > 0, we consider the notation X 2 3 .xj / D 1 C 2 e i  n  cos 2nx; n1

a

b

3

3

2

2

1

1 2

4

6

8

10

12

−1

−1

−2

−2

−3

−3

c

d

3.0

3.0

2.5

2.5

2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5 2

4

6

8

10

12

2

4

6

8

2

4

6

8

10

10

12

12

Fig. 1.3 The theta-functions plotted for q D 0:7: .a/ 1 .x; 0:7/, .b/ 2 .x; 0:7/, .c/ 3 .x; 0:7/ and .d/ 4 .x; 0:7/

10

1 Introduction

which is sometimes useful to be written as X 2 3 .xji / D 1 C 2 e  n  cos 2nx:

(1.5.4)

n1

Using Euler’s formula and the fact that sine is an odd function, we have the equivalent definition 1 X

3 .xj / D

2

q n e 2nxi D

nD1

1 X

2

e i  n  e 2nxi :

nD1

Similar considerations can be carried out for the other theta-functions. More properties of the theta-functions can be found in Chap. 1 of [86].

1.6 Some Useful Integrals When computing heat kernels, one often gets into integrals which can be reduced to one given by the following result. Proposition 1.6.1. We have Z

r

 b2 e 4a ; a r Z   2 2 e 4a ; e ay Ci y dy D a e

ay 2 Cby

dy D

8a 2 R ; b 2 R; 8a 2 R ; 2 R:

Proof. Completing the square and using the Gaussian integral yields Z e

ay 2 Cby

dy D e

b2 4a

Z

p

e



p 2 ayb=.2 a/

(1.6.5)

dy D e

b2 4a

1 p a

Z e

R

v2

(1.6.6) 2

e v d v D r dv D

p



 b2 e 4a : a

The second integral is obtained by formally replacing b by i . Next we shall present a complete proof of this result. Completing the square yields Z

e ay

2 Ci y

Z  p p 2 p 2 2  ayi =.2 a/  ui =.2 a/  4a 1 e e dy D e dyD e p du a Z 2 1 2 e z dz; (1.6.7) D p e  4a p a Im zD=.2 a/ 2

 4a

Z

where I m z denotes the imaginary part of the complex number z. The integral can be brought down to an integral on the real line as follows. Since the function p 2 e z is holomorphic, its integral on the contour f.R; 0/; .R; 0/; .R;  =.2 a//;

1.6 Some Useful Integrals

11

Fig. 1.4 The rectangular contour f.R; 0/;p a//; .R; 0/; .R;  =.2 p .R;  =.2 a//g

p 2 .R;  =.2 a//g vanishes; see Fig. 1.4. Since the limit of the integral of e z on the vertical edges of the aforementioned rectangle vanishes as R ! 1, it follows that Z Z p 2 z2 e dz D e x dx D : p Im zD=.2 a/

R

Substituting in (1.6.7) yields the desired result.



Chapter 2

A Brief Introduction to the Calculus of Variations

The Lagrangian and Hamiltonian formalisms will be useful in the following chapters when the heat kernel will be computed using the path integral and geometric variational methods. In the following we shall present a brief overview of the variational theory needed in the sequel.

2.1 Lagrangian Mechanics In classical mechanics a moving particle is completely described at any instance of time t by its position x.t/ and its velocity x.t/. P The position x belongs to the coordinate space, which is, in general, a Riemannian manifold with the metric defined by the kinetic energy. The space of the positions and velocities .x; x/ P is called the phase space, and it is identified with the tangent bundle TM of the coordinate space M . The pair .x; x/ P is called the state of the particle. A real-valued function defined on the tangent bundle L W TM ! R is called Lagrangian. In classical mechanics the usual Lagrangian is given by the difference between the kinetic energy and the potential energy of the particle: L.x; x/ P D K.x/ P  U.x/; P where K.x/ P D 12 niD1 xP i2 and U.x/ is usually a polynomial function of x. The trajectory of a particle x.t/ in the coordinate space is a curve parameterized by the time parameter t. The Lagrangian L.x; x; P t/ describes the dynamics of the particle, in the sense that the particle moves on a trajectory x.t/ such that the following action integral Z 

S.x0 ; x; / D

L.x.t/; x.t/; P t/ dt

(2.1.1)

0

is locally minimized under small variations of the path x.t/. We shall investigate this problem in the next few sections.

O. Calin et al., Heat Kernels for Elliptic and Sub-elliptic Operators, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-4995-1 2, c Springer Science+Business Media, LLC 2011 

13

14

2 A Brief Introduction to the Calculus of Variations

2.1.1 The First Variation For our study it would be sufficient to investigate the case of Lagrangians that do not depend explicitly on time t. Let x.t/ be a smooth curve joining x0 D x.0/ and x D x./. The action along x.t/ is given by 

 S D S x.t/ D

Z



  L x.s/; x.s/ P ds:

0

The main problem of Lagrangian formalism is formulated below: Given the fixed endpoints x0 and x, find the path x.t/ for which the functional   x.t/ ! S x.t/ (a) Has a “critical point” (b) Has a minimum Customarily, this problem is approached by considering a fixed path x.t/ and a variation x .t/ D x.t/ C .t/ with fixed endpoints x .0/ D x.0/ D x0 ;

x ./ D x./ D x0 :

The vector field .t/ defined on Œ0;  is an arbitrary variation vector field with homogeneous boundary conditions .0/ D ./ D 0:

(2.1.2)

Expanding S about the path x.t/ in powers of  yields    Z       @L @L S x .t/ D S x.t/ C  ; P C ;  dt @xP @x 0   2   2  2 Z   2 @ L @ L @ L  ; P  C ; P P C 2 ;  dt C O. 3 / C 2Š 0 @xP 2 @x@x P @x 2 2 2 ı S C O. 3 /; D S C  ıS C 2Š   The path x.t/ is a “critical point” for the action functional S x.t/ if and only if ıS D 0. Integrating by parts and using the boundary conditions (2.1.2) yields Z 0 D ıS D 0





    Z   @L @L d @L @L ; P C ;  dt D C ;  dt:  @xP @x dt @xP @x 0

2.1 Lagrangian Mechanics

15

Since the vector field  is arbitrary, we obtain the variational equations, which are the famous Euler–Lagrange equations @L d @L D ; dt @xP @x

(2.1.3)

where xP D .xP 1 ; : : : ; xP n / and x D .x1 ; : : : ; xn / denote the velocity and position of the particle, respectively. A solution of (2.1.3) is called a classical path and will play a central role in the rest of the book.

2.1.2 The Second Variation Assume x.t/ is a solution of the Euler–Lagrange equations; i.e., it is a classical path. If the second variation is positive definite along x.t/, i.e., P .t/i > 0; hı 2 S .t/;

8t 2 .0; /;

(2.1.4)

for any variation vector field .t/ with .0/ D ./ D 0, then the path x.t/ minimizes the action functional between x0 and x./. In order to have relation (2.1.4) satisfied, it suffices to prove that minfhı 2 S .t/; P .t/i; 8t 2 .0; /g > 0: 

Let  be a variation vector field for which the previous minimum is reached. Then  will satisfy the Euler–Lagrange equations for the associated Lagrangian L.; / P D hı 2 S .t/; P .t/i; which are given by d dt



  2

   2 d @ L @2 L @2 L @2 L @ L P C  2 C P D 0;  @xP 2 dt @x@xP @x @x@x P @x@xP

(2.1.5)

which is called the Jacobi equation. A solution .t/ of (2.1.5) is called a Jacobi vector field. If there is a value t1 2 .0; / such that a Jacobi vector field .t1 / D 0, then inequality (2.1.4) fails at t D t1 . A well-known result of variational calculus states that as long as the Jacobi vector field doesn’t vanish, the classical path is still minimizing the action. The classical path ceases to be a minimizer as soon as the Jacobi vector field vanishes the first time. The vanishing points of a Jacobi vector field are called conjugate points. This is equivalent to saying that the classical path is minimizing between two consecutive conjugate points and is no longer minimizing after that.

16

2 A Brief Introduction to the Calculus of Variations

2.1.3 Geometrical Interpretation Next we shall deal with the geometrical significance of the Jacobi vector fields. Consider the classical path starting at x0 and all the neighboring, classical paths x.p; t/ starting at the same point and parameterized by their initial momenta p; this is x.p; 0/ D x0 ; see Fig. 2.1. For the time being we neglect the fixed endpoint condition at t D . The separation between two classical paths is given by x.p C ; t/ D x.p; t/ C J.p; t/ C O. 2 /;   with J.p; t/ D Jij .p; t/, where Jij .p; t/ D

@xi .p; t/ ; @pj

i; j D 1; : : : ; n:

(2.1.6)

Since x.p; t/ are classical paths, the Euler–Lagrange equations are satisfied: @L d @L

D ; dt @xP r @xr

r D 1; : : : ; n:

Using the chain rule @L @xi @L @xP j @L D C @pk @xi @pk @xP j @pk @L @L P D Ji k C Jjk @xi @xP j and differentiating in the Euler–Lagrange equations yields  2  2     @L @2 L @ L @ L P d @2 L d   JPi k D 0: Ji k C Ji k C dt @xP r @xP i dt @xP r @xi @xr @xi @xP r @xi @xr @xP i (2.1.7)

Fig. 2.1 The Jacobi vector field along a classical path between two conjugate points

2.1 Lagrangian Mechanics

17

A standard result of ODEs shows that the second-order-system (2.1.7) together with the following 2n2 initial conditions @xi .p; 0/ D 0; @pk @xP i .p; 0/ D ıi k ; JPik .p; 0/ D @pk Jik .p; 0/ D

has a unique solution Jij .p; t/ 6D 0, for t 2 .0; /, with  > 0 small enough. Let p D x.0/ P be fixed. For any vector v D .v1 ; : : : ; vn / 6D 0, we can define the vector field .t/ D J.t/v along the classical path x.t/, by i .t/ D Ji k .t/vk . If there is a T > 0 such that .T / D 0, then det Ji k .T / D 0. The point x.T / is conjugate with x0 D x.0/, in the sense of the definition given in the previous section. This fact will be shown next. Multiplying (2.1.7) by vk and summing over k yields the equation  2      2 @ L i d @2 L d @2 L @2 L @ L i C P D 0; P C   dt @xP r @xP i dt @xP r @xi @xr @xi @xP r @xi @xr @xP i which is exactly the Jacobi equation (2.1.5). The vector field .t/ becomes a Jacobi vector field. Since Ji k is nondegenerate for t 2 .0; T /, the set of Jacobi vectors forms an n-dimensional space at every point along the path x.t/. The first point where one Jacobi vector (and hence all of them) vanishes is a conjugate point with x0 . The classical path x.t/ minimizes the action as long as it does not pass through a conjugate point. In the following we shall write the matrix J , which satisfies (2.1.7), in terms of the action S . Let x0 and x be two fixed endpoints and let x.t/ be a classical path such that x.0/ D x0 and x./ D x. The momentum along x.t/ at time t is denoted by p.x0 ; xI t/. Let e x .t/ be the reverted curve defined by e x .t/ D x.  t/ D x.tQ/. The curves e x .t/ and x.t/ have the same endpoints, x0 D e x ./ D x.0/ and x D e x .0/ D x./. The following relation between the momenta along the curves x.t/ and e x .t/ holds: p.x0 ; xI t/ D p.x; x0 I   t/ D p.x; x0 I tQ/: Using the well-known relation

@S @x

D p along x.t/ at instances t and  yields

@S.x0 ; x.t/I t/ D p.x0 ; x.t/I t/; @x.t/

@S.x0 ; xI / D p.x0 ; xI /: @x

A similar argument for the curve e x .t/ yields @S.x; x0 I tQ/ D p.x; x0 I tQ/ D p.x0 ; xI t/: @x0

18

2 A Brief Introduction to the Calculus of Variations

Differentiating with respect to x yields @S @p.x0 ; xI t/ 1 D D : @x@x0 @x J.p; x/ This means that

@S @x i @x0k

(2.1.8)

and Ji k are inverse matrices. Since at conjugate points

det Ji k D 0, it follows that at conjugate points D D det 

@S

D ˙1: @x@x0

One of the main properties of D is that it satisfies a continuity equation of the following type (see [54]): @D X @ C .xP k D/ D 0; @t @xk

(2.1.9)

k

which means that D can be interpreted as a density of paths. This fact will be useful in the future chapters that deal with path integration, van Vleck’s formula and the geometric method of computing heat kernels.

2.1.4 The Case of Riemannian Geometry P Consider the Lagrangian L.x; x/ P D 12 ni;j D1 gij .x/xP i xP j , with gij a positive definite and non-degenerate matrix at each point x 2 Rn . This Lagrangian can also be considered on the tangent bundle of an n-dimensional Riemannian manifold .M; g/. However, since we are studying local properties, we can make the simplifying assumption M D Rn . In this case the Euler–Lagrange equations become the familiar equations of geodesics xR k .t/ C

X

ijk .x.t//xP i .t/xP j .t/ D 0;

k D 1; : : : ; n:

i;j

The classical paths are called geodesics and satisfy the following local result. Theorem 2.1.1. Given a point x0 on a Riemannian manifold .M; g/, there is a neighborhood V of x0 such that for any x 2 V, there is a unique geodesic joining the points x0 and x. The aforementioned result does not necessarily hold globally for any Riemannian manifold. However, it holds on compact manifolds, and in general on metrically complete manifolds, as the Hopf–Rinov theorem states; see [79].

2.1 Lagrangian Mechanics

19

Moreover, any geodesic is locally minimizing the action functional. The distance on the Riemannian manifold .M; g/ is defined by d.A; B/ D inff`.x/ j x geodesic; x.0/ D A; x./ D Bg; where the length of x.t/ is Z `.x/ D 0



sX

gij .x.t//xP i .t/xP j .t/ dt:

i;j

The classical action in this case is given by the formula S.x0 ; xI t/ D

d 2 .x0 ; x/ : 2t

Along a geodesic we have rS D x.t/, P where r is the gradient in the metric g: g.U; rS/ D dS.U /; for any tangent vector field U ; see [24]. The Jacobi equation (2.1.5) on Riemannian manifolds takes the form   .t/ R D R .t/x.t/ P x.t/; P

(2.1.10)

where R is the Riemannian curvature tensor induced by the Levi–Civita connection D: R.U; V /W D DŒU;V  W  ŒDU ; DV W: If the manifold has constant curvature K, then   R.U; V /W D K g.W; U /V  g.W; V /U : In this case, under the additional hypotheses that x.t/ is unit speed and the Jacobi vector field  is normal to the geodesic x.t/, the Jacobi equation (2.1.10) can be written in the suggestive form .t/ R D K.t/;

(2.1.11)

with the initial condition .0/ D 0. Solving, we distinguish the following cases: (1) Euclidean case: K D 0, .t/ D ct, c constant. See Fig. 2.2a. (2) Elliptic case: K D k 2 > 0, .t/ D c sin.kt/. See Fig. 2.2b. (3) Hyperbolic case: K D k 2 < 0, .t/ D c sinh.kt/. See Fig. 2.2c. In cases (1) and (3) there are no conjugate points, while in case (2) there are infinitely many conjugate points to x.0/ that occur at tn D n=k, n D 1; 2; : : : :

20

2 A Brief Introduction to the Calculus of Variations

a

b

c

Fig. 2.2 .a/ Euclidean case: K D 0; .b/ elliptic case: K > 0; .c/ hyperbolic case: K < 0

This behavior, for instance, occurs on a sphere. In general, all manifolds in situation (2) are compact. Just for the record, we include here a generalization of this case. Recall the notation for the diameter of a manifold as diam.M / D supfd.p; q/I p; q 2 M g. Theorem 2.1.2 (Myers). Let .M; g/ be a complete, connected n-dimensional Riemannian manifold. If there is a positive number k > 0 such that Ric  .n  1/k 2 g;

(2.1.12)

where Ric denotes the Ricci tensor of M , then the following relations hold: (i) diam.M /  =k (ii) M is compact As a special case of the previous theorem, we have Corollary 2.1.3. If .M; g/ is a complete, connected n-dimensional Riemannian manifold with constant curvature K satisfying K  k 2 > 0, then (i) diam.M /  =k (ii) M is compact References for Riemannian geometry and its variational methods are the books [24, 79, 94].

2.1.5 Examples of Lagrangian Dynamics In the following examples we shall assume that the initial and final positions of the particle x0 D x.0/ and x D x./ are given and we shall determine its trajectory between these endpoints.

2.1 Lagrangian Mechanics

21

Example 2.1.4 (The natural Lagrangian). Let U.x/ be a smooth potential, and consider the Lagrangian L D 12 xP 2  U.x/. The Euler–Lagrange equation is @U : (2.1.13) x.t/ R D @x Given the boundary conditions x.0/ D x0 , x./ D x, (2.1.13) might not always have a unique solution, regardless of how close the boundary points x0 and x are: see [24]. If the potential is linear or quadratic, the aforementioned boundary value equation has a unique solution, and in this case the action is well defined. The Jacobi equation for the previous Lagrangian takes the form ˇ @2 U.x/ ˇˇ .t/ R D ˇ @x 2 ˇ

.t/:

(2.1.14)

x.t /

We note that if U.x/ is linear or quadratic in x, then the previous equation becomes similar to (2.1.11). Example 2.1.5 (The free particle). If the Lagrangian is L.x; P x/ D 12 jxj P 2 D P 2 1 P j , then the variational equation (2.1.13) is x.t/ R D 0, and the trajectories are j x 2 the straight lines  t x.t/ D x0 C x  x0 : (2.1.15)  The Jacobi equation (2.1.14) is .t/ R D 0. Using the initial conditions .0/ D 0, .0/ P D 1 yields the Jacobi vector field .t/ D t. The classical action is obtained by integrating the Lagrangian along the classical path: Z



S.x0 ; xI / D 0

.x  x0 /2 1 x  x0 2 dt D : 2  2

We recuperate formula (2.1.8) by taking the mixed derivative of the action @2 S 1 1 D D 6D 0: @x@x0  ./ Hence there are no conjugate points to x0 in this case. Next we shall check the continuity equation (2.1.9). We have D D 1t , xP k .t/ D

x k x0k , t

X @D @ 1 X @ @xk .vk D/ D C C @t @t t @xk n

kD1

n

kD1

and then

x k  x0k t2

!

1 1 D  2 C 2 D 0: t t Example 2.1.6 (Particle in constant gravitational field). The Lagrangian describing the dynamics of a free-falling particle under the action of a constant gravitational

22

2 A Brief Introduction to the Calculus of Variations

force is given by L.x; P x/ D 12 xP 2  kx, k > 0. The trajectory x.t/ satisfies Galileo’s equation xR D k with the solution k t k x.t/ D  t 2 C .x  x0 / C t C x0 : 2  2 Since the Jacobi equation is .t/ R D 0 with the Jacobi vector field .t/ D t 6D 0, for t > 0, there are no conjugate points along the trajectory. We leave the computation of the classical action as an instructive exercise to the reader. Example 2.1.7 (The linear oscillator). If L.x; P x/ D 12 xP 2  the Euler–Lagrange equation is x.t/ R D kx, with the solution

k 2 x , 2

k > 0, then

p p

sinh. kt/ x.t/ D x  x0 cosh. k/ C x0 cosh. kt/: p sinh. k/

p

This represents the trajectory of a particle under an elastic force F .x/ D kx. The Jacobi equation (2.1.14) can be R D k.t/, and the Jacobi vector field pwritten as .t/ is given by .t/ D p1 sinh. kt/ 6D 0, for t > 0. Then there are no conjugate k points to x0 along the classical path x.t/. A tedious computation shows that the classical action in this case is p p

2 k p .x C x02 / cosh. k /  2xx0 : S.x0 ; xI / D 2 sinh. k / Differentiating yields p @2 S 1 k D ; p D @x@x0 .t/ sinh. k / which is (2.1.8). Example 2.1.8 (The simple pendulum). The dynamics of a unit mass pendulum bob with unit length pendulum string under the action of the gravitational force is modeled by the Lagrangian L.x; P x/ D 12 xP 2  k.1  cos x/, k > 0. Its variational equation is given by the second-order differential equation x.t/ R D k sin x that cannot be solved using elementary functions. A solution using elliptic functions can be found in [24], p. 38. In the case of small oscillations the dynamics is approximated by the linearized pendulum equation x.t/ R D kx, with the solution p p p

sin. k t/ x.t/ D x  x0 cos. k / C x0 cos. k t/: p sin. k/ The Jacobi equation (2.1.14) becomes .t/ R D k.t/:

2.2 Hamiltonian Mechanics

23

p Under the standard initial conditions, the Jacobi vector field is .t/ D p1 sin. k t/. k p Hence the conjugate points to x0 will occur along x.t/ at instances tn D n= k, n D 1; 2; : : : . The classical action is p p p

2 k  … fn= kg: p .x C x02 / cos. k /  2xx0 ; S.x0 ; xI / D 2 sin. k / Example 2.1.9 (Particle in a constant electric field). The Lagrangian of a particle in a one-dimensional electric potential U.x/ D k=x, x > 0, is L.x; P x/ D 12 xP 2  2 k=x, k > 0. The trajectories satisfy the equation xR D k=x , which cannot be integrated using elementary functions. The same occurs for the Jacobi equation R D 

2k : x3

2.2 Hamiltonian Mechanics An alternate way of describing the dynamics of a particle is using the Hamiltonian function. A real-valued function defined on the cotangent bundle of the coordinate space T  M is called a Hamiltonian function. There is an intimate relationship between the Lagrangian and the Hamiltonian associated with a moving particle. The Hamiltonian associated with a Lagrangian is obtained using Legendre’s transform H.x; p/ D p xP  L.x; x; P t/; where xP is a function of the momentum p obtained by solving the equation pD

@L : @xP

(2.2.16)

For instance, if the Lagrangian L.x; x/ P D 12 xP 2  x m , then the Hamiltonian is H.x; p/ D

1 2 p C xm: 2

The procedure works vice versa, i.e., for a given Hamiltonian, the associated Lagrangian can be obtained by L.x; x; P t/ D p xP  H.x; p/; with the momentum p given by (2.2.16). The dynamics of the particle is described in the cotangent space by the Hamiltonian system of equations xP D @p H; pP D @x H:   A solution x.t/; p.t/ of the aforementioned system is called a bicharacteristic.

24

2 A Brief Introduction to the Calculus of Variations

If the Hamiltonian H does not depend explicitly on the time variable t, the system is said to be conservative, because in this case the Hamiltonian evaluated along  the above  solutions is constant. This can be verified by using the chain rule. If x.t/; p.t/ is a bicharacteristic curve, then  d  P @p H D 0: H x.t/; p.t/ D x.t/ P @x H C p.t/ dt The value of the Hamiltonian evaluated along the bicharacteristic is called the total energy of the particle. This is a first integral of motion. If .x.t/; p.t// is a bicharacteristic curve, then the component x.t/ is a solution of the Euler–Lagrange  equation.  Conversely, if x.t/ is a solution of the Euler–Lagrange equations, then x.t/; @L .t/ is a bicharacteristic curve, called the lift of x.t/. @xP

2.3 The Hamilton–Jacobi Equation The classical action associated with the Lagrangian L.x; x; P t/ is obtained by integrating the Lagrangian along the solution of the Euler–Lagrange equation Z    L x.t/; x.t/; P t dt: (2.3.17) Scl .x0 ; xI / D 0

This assumes that there is only one solution x.t/ satisfying the boundary conditions x.0/ D x0 and x./ D x. The classical action is a solution of the following Hamilton–Jacobi equation: @ Scl C H.x; @x Scl / D 0: @

(2.3.18)

Since the above equation is nonlinear, there might be more than one solution. In the case of a conservative system, if E D E.x0 ; x; / D H denotes the energy along the solution, the equation becomes @ Scl D E; @ with the solution Scl .x0 ; x; / D E.x0 ; x; / C c.x0 ; x/;

(2.3.19)

where c.x0 ; x/ D Scl .x0 ; x; 0/ is the initial condition. The action plays a very important role in the geometric methods of finding the heat kernel. To conclude our brief discussion, there are three ways of computing the action. 1. Starting from the Lagrangian. After solving the Euler–Lagrange equations, we integrate the Lagrangian along solutions using formula (2.3.17). This is a robust method and works as long as we are able to solve the Euler–Lagrange system of equations explicitly.

2.3 The Hamilton–Jacobi Equation

25

2. Solving the Hamilton–Jacobi equation. Substitute p D @x S in the expression of the Hamiltonian H and solve (2.3.18). Because it is nonlinear, there are no standard methods to solve the aforementioned equation. Depending on the problem, one may try different strategies. For instance, we can try to look for particular solutions of the type S.x; t/ D a.t/Cb.x/. Substituted in (2.3.18), the functions a.t/ and b.x/ satisfy the separable equation a0 .t/ C H.x; @x b/ D 0; so there is a constant C such that a0 .t/ D C H) a.t/ D C t C a.0/; H.x; @x b/ D C:

(2.3.20)

If the Hamiltonian H does not depend explicitly on the variable x, (2.3.20) can sometimes be reduced to an eikonal equation. For instance, if H D 12 jpj2 , then the function b.x/ satisfies j@x bj2 D C =2;

C > 0:

This equation has infinitely many solutions of the type r b.x/ D

v u n u2 X 2 .xi  xi0 /2 ; jx  x0 j D t C C

x0 2 Rn :

i D1

3. The conservative Hamiltonian case. If the Hamiltonian does not depend explicitly on the time parameter t, the action function is given by (2.3.19), where the energy should be expressed in terms of the endpoints x.0/ and x./. Next we shall work out an example. The aforementioned methods will be used to find the action function for the free particle case; see Example 2.1.5. Differentiating in (2.1.15) yields  1 xP D x./  x.0/ :  The Lagrangian along the solution is L.x; x/ P D

1 2 jx./  x.0/j2 ; xP D 2 2 2

and using (2.3.17) we obtain the action Z



S.x0 ; xI / D 0

jx  x0 j2 jx./  x.0/j2 jx./  x.0/j2 D : dt D 2 2 2 2

26

2 A Brief Introduction to the Calculus of Variations

If we try to solve the Hamilton–Jacobi equation, using @x S D p D (2.3.18) becomes

@L @xP

D x, P

@ @ 1 1 jx  x0 j2 2 D0” ” S C jx.t/j P S D @ 2 @ 2 2   @ @ jx  x0 j2 : SD @ @ 2 Integrating yields the aforementioned relation for the action. The reader can find more advanced topics on the calculus of variations in [6, 24, 52].

Chapter 3

The Geometric Method

This chapter deals with a construction of heat kernels from the geometric point of view. Each operator will be associated with a geometry. Investigating the geodesic flow in this geometry, one can describe the heat kernels for a large family of operators. The idea behind this method is that the heat flow propagates along the geodesics of the associated geometry. The “density” of the heat flow is described by a volume function that satisfies a transport equation which is an analog of the continuity equation from fluid dynamics. This corresponds to the density of paths given by the van Vleck determinant in the path integral approach. This method works for elliptic operators with or without potentials or linear terms. The method can be modified to work even in the case of sub-elliptic operators, as the reader will become familiar with in Chaps. 9 and 10. This method was initially applied for the Heisenberg Laplacian; see, for instance [28].

3.1 Heat Kernel for L D

1 2

Pn

i;j D1

aij @xi @xj

P Consider the elliptic differential operator L D 12 ni;j D1 aij @xi @xj . Since the matrix aij is symmetric, nondegenerate and positive definite at each point, then .Rn ; aij / becomes a Riemannian space, where .aij /1 D .aij /. The study of the geometry of this space provides a method for finding the heat kernel of the operator L. The heat kernel models the physical phenomenon of heat propagation from a point x0 to another point x within time t. The main idea is that the heat flows along the geodesics of the space .Rn ; aij / from the initial point x0 to the final point x. For the sake of simplicity, we shall assume in Theorems 3.1.1 and 3.4.3 that the Riemannian space .Rn ; aij / has the property that any two points can be joined by a unique geodesic, i.e., the space is geodesically complete, and there are no conjugate or cut points along the geodesics. This condition is always satisfied locally on a Riemannian manifold, so the aforementioned theorems provide a local expression for the heat kernels. However, in some cases, for instance in the case of spaces with negative or zero curvature, this property is globally satisfied, and hence the expression for the heat kernels is globally defined.

O. Calin et al., Heat Kernels for Elliptic and Sub-elliptic Operators, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-4995-1 3, c Springer Science+Business Media, LLC 2011 

27

28

3 The Geometric Method

The case when the cut locus of x0 is not empty, i.e., there is more than one geodesic joining the points x0 and x, the analysis is more complicated and shall be treated separately. The geometric method described in this chapter is based on finding geodesics. Obtaining the geodesics. The geodesics of the Riemannian space .Rn ; aij / can be obtained in two ways. Consider the following Hamiltonian function obtained by taking the principal symbol of the operator L: H.x; p/ D

n 1 X aij pi pj : 2

(3.1.1)

i;j D1

One way to define the geodesics is by considering the x-component of the bicharacteristic curve that solves the Hamiltonian system xP D Hp ; pP D Hx with the boundary conditions x.0/ D x0 ;

x.t/ D x:

The solution will be a geodesic joining the points x0 and x. For a large class of ODE systems, the solution of the previous boundary value problem is unique. The alternate approach for finding the geodesics is by solving the Euler– Lagrange system for the Lagrangian associated with the Hamiltonian (3.1.1). Assuming that we were able to find the geodesic x.s/ in one way or another, we can measure its length by the formula v Z tuX u n t `.x/ D aij .x.t//x i .t/x j .t/ dtI 0

i;j D1

Denote by d.x0 ; x/ the Riemannian distance between the points x0 and x measured in the metric aij . This is the length of the shortest geodesic x.s/ which joins x0 D x.0/ with x D x.t/. This length is independent of the parameter t, in the sense that it is invariant when the geodesic changes its parameterization. One may show that the associated action is S.x0 ; xI t/ D

d 2 .x0 ; x/ I 2t

see, for instance, [24]. The aforementioned action satisfies the Hamilton–Jacobi equation @t S C H.rS / D 0I (3.1.2)

3.1 Heat Kernel for L D

1 2

Pn

i;j D1 aij @xi @xj

29

where H is the Hamiltonian (3.1.1). There is an alternate way of finding the action by integrating the Lagrangian along the geodesic Z

t

S.x0 ; xI t/ D 0

 1  L x.s/; x.s/ P ds; 2

where x W Œ0; t ! Rn is the geodesic which satisfies x.0/ D x0 and x.t/ D x. There is an intimate relationship between the geometric method and the path integration technique. In Chap. 7 the previous action is called the classical action and is denoted by Scl . It will be shown that the heat kernel can be represented as a path integral by Z K.x0 ; xI t/ D

Px0 ;xIt

e S.;t / d m./ D e Scl .x0 ;xIt /

Z

e S. P0;0It

;t /

d m. /:

R We shall see that in several cases the path integral P0;0It e S. ;t / d m. / is a function of t only. In this case the heat kernel takes the friendly form K.x0 ; xI t/ D V .t/e Scl .x0 ;xIt / ;

t > 0;

(3.1.3)

so we may say that the geometric method is a particular case of the path integral method. More precisely, we have the following result. P Theorem 3.1.1. Consider the operator L D 12 ni;j D1 aij @xi @xj and assume that L.Scl / is a function of t only. If there is only one geodesic joining the points x0 and x within time t, then the heat kernel of L is given by (3.1.3), where V .t/ satisfies the following transport equation: V 0 .t/ C V .t/L.Scl / D 0;

(3.1.4)

with the initial condition Z

V .0/ D lim

t &0

1 e Scl .x;x0 I0/ dx

:

(3.1.5)

Proof. Let S D Scl .x0 ; xI t/ denote the classical action. Since   @xi @xj e S D @xi  e S @xj S D e S .@xi S @xj S  @xi @xj S /; then L.e S / D

n 1 X aij @xi @xj e S D e S .H.rS /  L.S // : 2 i;j D1

(3.1.6)

30

3 The Geometric Method

On the other hand,   @t e S V .t/ D @t e S V .t/ C e S @t V .t/ D e S .@t V .t/  V .t/@t S / :

(3.1.7)

Subtracting (3.1.6) and (3.1.7) yields       .@t  L/ e S V .t/ D @t e S V .t/  L e S V .t/   D e S @t V C V .t/ L.S /  V .t/ .@t S C H.rS // ƒ‚ … „ D0

S

De D 0;

.@t V C V .t/ L.S //

where we used the Hamilton–Jacobi equation (3.1.2) and the transport equation (3.1.4). The solution V .t/ of the transport equation is unique up to a multiplicative constant. Relation (3.1.5) is fixing this constant. The reason behind relation (3.1.5) is the normalization condition for the heat kernel Z K.x; x0 I t/ dx D 1:  Remark 3.1.2. Since the differential operator L is quadratic, the term L.S / does not depend on x in the cases when the action S depends on x linearly or quadratically. Formula (3.1.3) resembles van Vleck’s formula with van Vleck’s determinant V .t/. The function V .t/ shows how the geodesics spread away from the point x0 as t increases. For this reason we shall refer to V as the volume function.

3.2 Interpretation of the Volume Function One of the well-known conservation equations in fluid dynamics is the continuity equation @ C div. vk / D 0; @t where v D .v1 ; : : : ; vn / denotes the velocity of the fluid and  is the fluid density. If the fluid follows the direction of the geodesic flow of a Riemannian manifold, then the velocity of the flow is related to the action by v D rScl :

3.3 The Operator L D

1 2

Pn

2 iD1 bi @xi

31

Assuming the fluid is homogeneous, i.e., it has the same density .t/ at each point, the aforementioned equation becomes 0 .t/ C .t/divrScl D 0: Substituting V .t/ D

p

.t/, we get V 0 .t/ C V .t/LScl D 0;

where L D 12 . This is exactly the transport equation (3.1.4) in the case of the Laplace–Beltrami operator. Hence the square of the volume function describes the “density” of the geodesic flow, i.e., the way the geodesics spread away from a given geodesic. We also note the relation with the Jacobi vector field emphasized in Chap. 2. In the following we shall deal with explicit computations in a few particular cases.

3.3 The Operator L D Consider the operator L D Hamiltonian function

1 2

1 2

Pn

i D1

Pn

i D1

bi @x2 i

bi @2xi , with bi > 0 constants. Associate the 1X bi pi2 : 2 n

H.p/ D

i D1

Solving the associated Hamiltonian system xP i .s/ D bi pi ; pPi .s/ D 0; x.0/ D x0 ;

x.t/ D x;

yields xi .s/ D bi pi s C x0 D

xi  xi0 s C xi0 ; t

i D 1; : : : ; n;

where x.t/ D x D .x1 ; : : : ; xn /, x.0/ D x 0 D .x10 ; : : : ; xn0 /, and the pi are constants. The associated Lagrangian is 1X 1 2 xP ; 2 bi i n

L.x; x/ P D

i D1

32

3 The Geometric Method

and the action becomes Z

t

S.x0 ; xI t/ D 0

Z t

n   1X 1 L x.s/; x.s/ P ds D 2 bi

0

i D1

xi  xi0 t

2 ds

n X .xi  xi0 /2 : D 2bi t i D1

Since n 1X bj @2xj LS D 2

n X .xi  xi0 /2 2bi t

j D1

!

n ; 2t

D

i D1

the transport equation becomes V 0 .t/ C

n V .t/ D 0: 2t

Integrating yields

V .t/ D ct n=2 : P By Theorem 3.1.1, the heat kernel of 12 niD1 bi @2xi is K.x0 ; xI t/ D ct

n=2 

Pn

e

2

i D1

.xi xi0 / 2bi t

:

The constant c is determined from the relation Z K.x0 ; xI t/ dx D 1;

(3.3.8)

Rn

by using the well-known formula Z

1

e 2

Pn

i;j D1

Mij ui uj

Rn

In our case ui D xi  xi0 and Mij D grating in (3.3.8) yields Z 1D Rn

D ct

K.x0 ; xI t/ dx D ct

n=2

d u1    d un D

1 ı , bi t ij

n=2

.2/n=2 : .det M /1=2

with det M D t n .

Z e

1 2

Pn

i D1

n Y .2/n=2 n=2 D c.2/ bi  1=2 t

Qn1 n

i D1 bi

i D1

1 i D1 bi / .

2

.xi xi0 /

Rn

Qn

bi t

!1=2 :

dx1    dxn

Inte-

3.4 The Generalized Transport Equation

33

Hence n Y

c D .2/n=2

!1=2 bi

:

i D1

We have arrived at the following result. Theorem 3.3.1. Let bi > 0. Then the heat kernel of L D K.x; x0 I t/ D .2 t/

n=2

n Y

!1=2 bi

e



1 2

Pn

2 i D1 bi @xi

is given by

2

Pn

.xi xi0 /

i D1

2bi t

;

t > 0:

i D1

When bi D 1, we obtain the familiar formula for the heat kernel of the Laplacian. Corollary 3.3.2. The heat kernel for L D K.x; x0 I t/ D .2 t/

1 2

n=2 

e

Pn

2 i D1 @xi

Pn

i D1

is given by 2

.xi xi0 / 2t

;

t > 0:

3.4 The Generalized Transport Equation In Theorem 3.1.1 we had assumed that L.Scl / depends on t only. In this case the transport equation (3.1.4) can easily be solved by integration. However, there are some cases when L.Scl / depends on both variables t and x. In these cases the transport equation is replaced by the generalized transport equation. This equation describes how the heat density evolves along the geodesics. We shall start with an example. One of the operators for which L.Scl / depends on both x and t is L D 12 x@2x . The following discussion applies to the domain fxI x > 0g, where the operator is elliptic. The associated Hamiltonian function is H.x; p/ D

1 2 xp : 2

The Hamiltonian system is xP D Hp D xp; 1 pP D Hx D  p 2 ; 2 with boundary conditions x.0/ D x0 , x.t/ D x. First we solve for p. Integrating in the equation 

pP 1 D 2 p 2

(3.4.9)

34

3 The Geometric Method

yields pD

1 ; s0 C 12 s

with s0 2 R. Substituting in the first Hamiltonian equation, we get 1 xP H) ln x.s/ D D x s0 C 12 s and hence

 1 2 ds D ln s0 C s C C; 2 s0 C 12 s

Z

 1 2 x.s/ D e C s0 C s : 2

It is worth noting that x.s/ > 0 for s > 0 and hence only points on the positive semiaxis can be joined by a geodesic, i.e., no heat propagates in the negative semi-axis, a fact which agrees with the domain fxI x > 0g considered before. Given the endpoints x0 ; x > 0, we shall determine the unique constants C and s0 such that x0 D x.0/ and x D x.t/. This will show that the geodesic between x0 and x within time t is unique. Setting s D 0 and s D t, we arrive at the system x0 D e C s02 ;  1 2 C s0 C t : xDe 2 The elimination method provides s0 D

t p ; 2. x=x0  1/

eC D

p 4 .x C x0  xx0 /: 2 t

(3.4.10)

The Lagrangian associated with the Hamiltonian (3.4.9) is obtained by applying the Legendre transform L.x; x/ P D p xP  H D D

1 xP 2 xP xP  x 2 x 2 x

1 xP 2 : 2 x

Integrating along the geodesic and using (3.4.10), we obtain the classical action Z Scl .x0 ; xI t/ D 0

D

t

1 xP 2 .s/ ds D 2 x.s/

Z

t 0

p 2 .x C x0  xx0 /: t

2   1 e C s0 C 12 s 1 ds D e C t   2 2 e C s0 C 1 s 2 2

3.4 The Generalized Transport Equation

35

A straightforward computation shows L.Scl / D

1 2 1  x0 1=2 : x@x Scl D 2 4t x

Since the function V also depends on x, the hypothesis of Theorem 3.1.1 is not satisfied, and therefore the transport equation (3.1.4) does not hold. In the following we shall find the equation satisfied by the volume function V .t; x/. This will be called the generalized transport equation. Assume the heat kernel has the form K D V .t; x/e Scl . Since @x K D e Scl .@x V  V @x Scl /;   @2x K D e Scl @2x V  2@x Scl @x V C V .@x Scl /2  V @2x Scl ; @t K D e Scl .@t V  @t Scl V / ; we obtain 1 LK D @t K  x@2x K (2 De

! !) 1 1 2 2 2 @t V V @t Scl C x.Sx /  x @x V 2@x Scl @x V V @x Scl 2 2 „ ƒ‚ …

Scl

D e Scl



D0

  1  2 2 @t V  x @x V  2@x Scl @x V  V @x Scl ; 2

where we used that Scl satisfies the Hamilton–Jacobi equation 1 @t Scl C x.Sx /2 D 0: 2 Since LK D 0 for any t > 0, it follows that V .t; x/ satisfies the following generalized transport equation:  1  @t V  x @2x V  2@x Scl @x V  V @2x Scl D 0: 2

(3.4.11)

An explicit computation of the coefficients of @x V and V in the above equation yields r 1 2 1 1 x0 p V D 0: (3.4.12) @t V  x@x V C .2x  xx0 /@x V C 2 t 4t x This equation does not have a straightforward solution. The function V should be looked for as a product between an exponential term and a modified Bessel function, as in (8.20.79), where the operator is associated with a squared Bessel process.

36

3 The Geometric Method

However, an easy explicit solution can be obtained if x0 D 0. In this particular case it is not hard to check that a solution is cx V .t; x/ D 2 ; t with c constant. In this case the classical action is Scl .0; xI t/ D 2t x. By Theorem 3.1.1, the heat kernel for 12 x@2x from the origin is K.0; xI t/ D V .t; x/e Scl D

cx  2 x e t : t2

The constant c is determined from the condition Z K.0; xI t/ dx D 1: x>0

Since integration by parts shows that Z 1 c cx  2 x e t dx D ; 2 t 4 0 it follows that c D 4. If x0 D 0, then s0 D 0, and e c D 4x=t 2 , and the geodesics joining x0 D 0 and x at time t is x.s/ D xs 2 =t 2 . Proposition 3.4.1. The heat kernel for 12 x@2x from the origin is given by K.0; xI t/ D

4x  2 x e t ; t2

t > 0; x > 0:

Proof. We have shown that 12 x@2x K D 0 for t > 0. We still need to check that lim K.0; xI t/ D ı0 in the distribution sense. For any test function , we have t &0

Z

1

lim K.0; xI t/./ D lim

t &0

K.0; xI t/.x/ dx

t &0 0

Z

1

D lim

t &0 0

Z

1

D lim

t &0 0

Z

D .0/ „0

 4x  2 x 2x t e .x/ dx make u D t2 t   Z 1 tu t tu 4 tu u u  ue  du e d u D lim 2 t 2 2 2 t &0 0 2

1

ue u d u D .0/ D ı0 ./: ƒ‚ … D1

 If x0 > 0, x > 0, then the heat travels in infinitely many ways between x0 and x since it is reflected at the wall x D 0; the kernel will be given by a series in this case.

3.4 The Generalized Transport Equation

37

Remark 3.4.2. The change of variable x D r 2 transforms the operator 12 x@2x into an operator that looks just like a two-dimensional Bessel operator with a changed sign  1 2 1 2 1 @  @r : x@ D 2 x 8 r r The above calculations also hold in the general case. This is given by the following result. P Theorem 3.4.3. Consider the operator L D 12 ni;j D1 aij @xi @xj with .aij / a symmetric, non-degenerate matrix. Assume L.Scl / depends on both variables x and t. If there is a unique geodesic joining the points x0 and x within time t, then the heat kernel of L is given by K.x0 ; xI t/ D V .t; x/e Scl .x0 ;xIt / ; where V .t/ satisfies the following generalized transport equation: X aij @xi S @xj V  LV C V LS D 0; @t V C

(3.4.13)

(3.4.14)

i;j

Z

with lim

t &0

V .t; x/e Scl .x;x0 It / dx D 1:

(3.4.15)

Proof. To simplify notations, we shall denote the classical action by S D Scl .x0 ; xI t/. Differentiating in (3.4.13) yields  @xj @xi K D e S V @xi S @xj S C @xi @xj V  V @xi @xj S  .@xi S @xj V C @xi V @xj S / ; LK D e S fVH.rS / C LV  V LS  a.rS; rV /g ; @t K D e S f@t V  V @t S g; where 1X aij @xi S @xj S; 2 X a.rS; rV / D aij @xi S @xj V: H.rS / D

Since S satisfies the Hamilton–Jacobi equation @t S C H.rS / D 0; and V satisfies the generalized transport equation (3.4.14), we get .@t  L/K D e S f@t V C a.rS; rV /  LV C V LS g D 0:



Remark 3.4.4. If V does not depend on x, the generalized transport equation (3.4.14) becomes the usual transport equation (3.1.4).

38

3 The Geometric Method

The first two terms of the generalized transport equation (3.4.11) can be written as a derivative along the classical path x.t/: X @V  d  @V @V xP k D V t; x.t/ D C C a.rV; x/ P dt @t @xk @t @V C a.rV; rS /: D @t Hence we obtain another form of the generalized transport equation,  d  V t; x.t/ C V LS D LV: dt

3.5 Solving the Generalized Transport Equation The generalized transport equation (3.4.14) is in general hard to solve explicitly. Its solution V .t; x/ might be represented as an integral formula rather than being an elementary function. The following computation holds in the case of an elliptic operator and uses a series expansion for t small. A parametrix expansion can also be found in [84]. We shall consider V .k/ .t; x0 ; x/ D .2 t/n=2



0 .x0 ; x/

C

1 .x0 ; x/t

CC

k .x0 ; x/t

k

 ;

which approximates the volume function locally for t small, as a correction to the Euclidean volume element. Then K .k/ .x0 ; xI t/ D Vk .t; x0 ; x/e Scl .x0 ;xIt / is an approximation of the heat kernel K.x0 ; xI t/ D K .k/ .x0 ; xI t/ C O.t k /; We need to solve for

m

t  0:

recursively for small t. A computation shows

 n @t V .k/ D .2 t/n=2  2t

0C

 n 1 2

1

 n C t 2 2

2

 n    Ct k1 k  k : 2 (3.5.16)

Since the classical action on a Riemannian manifold is related to the Riemannian distance by d 2 .x0 ; x/ ; Scl .x0 ; xI t/ D 2t then rScl D

1 rd 2 .x0 ; x/. 2t

Hence

a.rScl ; rV .k/ / D .2 t/n=2

k 1 X  2 a rd .x0 ; x/; r 2t j D0

 j

t j:

(3.5.17)

3.5 Solving the Generalized Transport Equation

We also have LV

.k/

n=2

D .2 t/

39

k X

L

j .x0 ; x/ t

j

(3.5.18)

j D0

and V .k/ LScl D .2 t/n=2

k X 1  2 L d .x0 ; x/ 2t

j .x0 ; x/t

j

:

(3.5.19)

j D0

Using (3.5.16)–(3.5.19) and equating the coefficients of t j with 0 in the generalized transport equation for V .k/ ,   @t V .k/ C a rS; rV .k/  LV .k/ C V .k/ LS D O.t k /; yields the following system in the unknown functions   2 Ld  n  2  Ld  n C 2   2 Ld  n C 4 

Ld 2  n C 2k

0;

1; : : : :

0

C a.rd 2 ; r

0/

D 0;

1

C a.rd 2 ; r

1/

D 2L

0;

2

C a.rd 2 ; r

2/

D 2L

1;

:: :

 k

D

C a.rd 2 ; r

k/

(3.5.20)

:: :

D 2L

k1 ;

together with the initial conditions j .x0 ; x0 / D 0, j D 1; : : : ; k. In general, solving the above system is as hard as solving the generalized transport equation. The system can be solved only in simple cases, as we shall see in the next example.   Example 3.5.1. Let L D 12 @2x . Then d 2 .x0 ; x/ D .x  x0 /2 , L d 2 .x0 ; x/ D 1, and rd 2 .x0 ; x/ D 2.x  x0 /. The first equation of the above system becomes 2.x  x0 / with the solution

0

1

C 2.x  x0 /

with the solution 1 1 .x0 /

D 0;

(3.5.21)

D c0 , a constant. Then the second equation becomes 2

because

0 0

D

0 1

D 0;

c1 D 0; x  x0

D 0. Inductively we obtain all

j

D 0, j  1.

In the following we shall discuss a few more examples.

40

3 The Geometric Method

3.6 The Operator L D 12 y m .@x2 C @y2 /, m 2 N We shall discuss the cases m D 0; 1 and 2. Case m D 0. In this case the operator is the usual two-dimensional Laplacian L D 1 2 .@ C @2y / with the heat kernel 2 x K.x0 ; xI t/ D

.xx0 /2 C.yy0 /2 1 2t : e .2 t/

Case m D 1. The operator becomes L D 12 y.@2x C @2y /. We shall consider it in the domain f.x; y/I y > 0g where the operator is elliptic. The Hamiltonian function in this case is 1 (3.6.22) H D y.p12 C p22 /; 2 and the associated Lagrangian is given by LD

1 2 .xP C yP 2 /: 2y

We are interested in the geodesics and the action from .x0 ; y0 / to .x; y/ within time t. From the Hamiltonian system we have xP D Hp1 D yp1 ; yP ; y pP1 D Hx D 0 H) p1 D k constant; 1 1 pP2 D Hy D p12 C p22 : 2 2 yP D Hp2 D yp2 H) p2 D

Since the Hamiltonian function H does not depend explicitly on the parameter s, it is preserved along the solutions; i.e., there is a constant C > 0, which depends on the boundary points .x0 ; y0 /; .x; y/ and t, such that H.x; y; p1 ; p2 / D

1 2 C : 2

Using the expression of the Hamiltonian (3.6.22) yields the following equation in y:  yP 2 k2 C 2 D C 2: y Solving for yP yields yP 2 D C 2 y  k 2 y 2 :

(3.6.23)

3.6 The Operator L D 12 y m .@2x C @2y /, m 2 N

41

Completing the above to a square, we obtain ( )  4 2 2 1 C C  y yP 2 D k 2 : 4k 4 2 k2 With the substitution u D y  ( uP D k 2

2

1 C2 , 2 k2

the previous equation becomes

C2  u2 2k 2

2 )

r H) uP D ˙k

C2  u2 : 2k 2

Separating and integrating yields Z

u.s/

Z

du

s

k” 0  u2  2  2k 2k u.s/ D arcsin u.0/ ˙ ks ” arcsin C2 C2 C2 u.s/ D sin. ˙ ks/; 2k 2 u0

where

q

D˙

C2 2k 2 2



2k 2  D arcsin u.0/ : C2

Going back to the variable y, we obtain y.s/ D

1 C2 .1 C sin. ˙ ks// : 2 k2

Integrating in the first Hamiltonian equation xP D ky yields Z s x.s/ D x0 C k y 0  cos. ˙ ks/ k C2 : s D x0 C 2 k2 ˙k

(3.6.24)

(3.6.25)

Sign convention. If we choose the “plus” sign in front of k, it means that y.s/ is increasing, and hence y > y0 . If we choose the “negative” sign in front of k, it means that y.s/ is decreasing, and hence y < y0 . Relations (3.6.25) and (3.6.24) provide the equations of the geodesics. In the following we shall eliminate their trigonometric part. Rewrite the equations as k C2 1 C2 sD cos. ˙ ks/; 2 2 k ˙2 k 2 1 C2 1 C2 D sin. ˙ ks/: y.s/  2 k2 2 k2

x.s/  x0 

42

3 The Geometric Method

Summing the squares yields the following implicit equation:  2  2 k C2 1 C2 1 C4 x.s/  x0  s C y.s/  D : 2 k2 2 k2 4 k4

(3.6.26)

Making s D 0 yields 2  1 C4 1 C2 D ” y 2 k2 4 k4  C2 y0 y0  2 D 0: k Since y0 6D 0, it follows that

C2 D y0 : k2

(3.6.27)

Then (3.6.26) can be rewritten as 2  2  k 1 1 x.s/  x0  y0 s C y.s/  y0 D y02 ” 2 2 4 2    k x.s/  x0  y0 s C y.s/ y.s/  y0 D 0: 2

(3.6.28)

From (3.6.28) it follows that on the half-plane f.x; y/I y > 0g the solution y.s/ is a decreasing function of s. If by contradiction we assume that y.s/ is increasing, then 0 < y0 < y.s/ and hence y.y  y0 / > 0 and (3.6.28) cannot hold. Hence the sign in front of k in formulas (3.6.24)–(3.6.25) must be negative. A particular family of geodesics is obtained when both terms of (3.6.28) vanish: k y0 s 2 x  x0 s; D x0 C t y.s/ D y0 : x.s/ D x0 C

In this case the geodesics are lines and the action is   S .x0 ; y0 /; .x; y0 /I t D

Z Z

t 0 t

D 0

1 P 2 / ds .x.s/ P 2 C y.s/ 2y.s/ 1 .x  x0 /2 1  x  x0 2 ds D : 2y0 t 2y0 t

3.6 The Operator L D 12 y m .@2x C @2y /, m 2 N

Since LS D

43

 1 .x  x0 /2 1 1 D ; y.@2x C @2y / 2 2y0 t 2t

the transport equation is 1 V .t/ D 0; 2t c with the solution V .t/ D t 1=2 , c a constant. Then the heat kernel between .x0 ; y0 / and .x; y0 / is given by Theorem 3.1.1: V 0 .t/ C

  c  1 .xx0 /2 K .x0 ; y0 /; .x; y0 /I t D 1=2 e 2y0 t ; t Finding the constant c. Z 1D D

K dx D c t 1=2

c

Z

t 1=2

e

1  2y

0

.xx0 /2 t

t > 0:

dx

 .2y0 t/1=2 D c.2y0 /1=2 H) c D

1 : .2y0 /1=2

Hence   K .x0 ; y0 /; .x; y0 /I t D

2 1 .xx0 / 1  2y t 0 e ; .2y0 t/1=2

t > 0:

The general case. From the Hamiltonian equation xP D ky and y > 0, it follows that if k > 0, then x.s/ is increasing and hence x0 < x. And if k < 0, then x.s/ is decreasing and then x0 < x. Making s D t in (3.6.28), we obtain  2   k x  x0  y0 t C y y  y0 D 0; 2 and then solving for k, we get

kD

8 p   2 ˆ < y0 t x  x0 C y.y0  y/ ;

if x > x0 I

p   x  x0  y.y0  y/ ;

if x < x0 :

ˆ :

2 y0 t

We note that y0  y always, since y.s/ is decreasing. Finding the action between .x0 ; y0 / and .x; y/ within time t. Since the Lagrangian along the geodesics is 1 2 1 2 2 .xP C yP 2 / D .y p1 C y 2 p22 / 2y 2y 1 1 D y.p12 C p22 / D H D C 2 ; 2 2

LD

44

3 The Geometric Method

integrating yields the action   S .x0 ; y0 /; .x; y/I t D

Z

t

1 2 1 C2 2 1 k t D y0 tk 2 C tD 2 2 2k 2 0 8 p   2 < y2 t x  x0 C y.y0  y/ ; if x > x0 I 0 D : 2 x  x  py.y  y/2 ; if x < x : 0 0 0 y0 t LD

Since LS D 12 y.@2x C @2y /S depends on t, x and y, the heat kernel is given in this case by Theorem 3.4.3: 8  2 p ˆ  2 xx0 C y.y0 y/ ; if x > x0 I  < V .t; x; y/e y0 t  K .x0 ; y0 /; .x; y/I y D   2 p ˆ :  2 xx0  y.y0 y/ V .t; x; y/e y0 t ; if x < x0 ; where V .t; x; y/ satisfies the generalized transport equation (3.4.14). Case m D 2. The operator becomes L D 12 y 2 .@2x C @2y /, which is the Laplace– Beltrami operator on the upperhalf-plane U D f.x; y/I y > 0g, with the metric ds2 D y12 .dx2 C dy 2 /. The Hamiltonian function is H D

1 2 2 y .p1 C p22 /; 2

(3.6.29)

with the associated Lagrangian Any two points of U can be joined by a unique geodesic, which is either a vertical radius or a half-circle perpendicular on the line fy D 0g. Hence, by Theorem 3.4.3, the heat kernel can be represented as a product     Scl .x0 ;y0 /;.x;y/It : (3.6.30) K .x0 ; y0 /; .x; y/I t D V .t; x; y/e Let dh denote the hyperbolic distance on the upperhalf-plane. If the points A.x0 ; y0 / and B.x; y/ belong to the same vertical line, then the distance between them is ˇ AM ˇ ˇ y ˇ ˇ ˇ ˇ ˇ ˇ 0ˇ ˇ ˇ dh .A; B/ D ˇ ln ˇ D ˇ ln ˇ D ˇ ln y  ln y0 ˇI BM y see Fig. 3.1a. Otherwise, ˇ BN=BM ˇ ˇ A0 M  NB 0 ˇ ˇ .K  x C r/.x  K C r/ ˇ 0 ˇ ˇ ˇ ˇ ˇ ˇ dh .A; B/ D ˇ ln ˇ D ˇ ln ˇ; ˇ D ˇ ln AN=AM AA0  BB 0 y0 y where r 2 D y02 C .K  x0 /2 ; KD

1 .x  x0 /2 C y 2  y02 I 2 x  x0

3.7 Heat Kernels on Curved Spaces

a

45

b A(xo,yo ) B(x,y)

N(-r,0)

A(xo,0)

O(K,0)

B (x,0)

M(r,0)

Fig. 3.1 The hyperbolic distance dh .A; B/. (a) The case when A and B are on a vertical direction; (b) A and B are on a non-vertical direction

see Fig. 3.1b. Since dh is the associated Riemannian distance on the upper space U, we have     dh2 .x0 ; y0 /; .x; y/ Scl .x0 ; y0 /; .x; y/I t D : 2t Since LScl does not depend on t only, the transport equation is messy and the heat kernel in this case is hard to find in the product form (3.6.30). However, McKean [89] found the integral representation p Z 1 s2 2 se  2t t =2 e p ds; (3.6.31) .2 t/3=2 cosh s  cosh     where  D dh .x0 ; y0 /; .x; y/ . It is interesting for further investigations to clarify the relationship between the integral representation (3.6.31) and the product formula (3.6.30). As we shall see in the next section, this relation is obvious in the case of a three-dimensional hyperbolic space.   K .x0 ; y0 /; .x; y/I t D

3.7 Heat Kernels on Curved Spaces Let g be the Laplace–Beltrami operator on a Riemannian space .M; gij / of dimension n. Let R D g ij Rij be the Ricci scalar curvature of the space, which will be assumed constant. Let d.x0 ; x/ denote the Riemannian distance between the points x0 and x. The heat kernel for the elliptic operator g is given by K.x0 ; x; t/ D where

d 2 .x0 ;x/ 1 g.x/1=4 D 1=2 g.x0 /1=4 e Rt e  4t ; n=2 .4 t/

2  @ Scl D D det  @x0 @x

is the van Vleck determinant; see Schulman [102], Chap. 24.

(3.7.32)

46

3 The Geometric Method

Applying the aforementioned formula for the classical spaces with constant curvatures 0; 1; 1, we arrive at the following classical results. The three-dimensional hyperbolic space. The Laplace–Beltrami operator on the three-dimensional hyperbolic space, i.e., the domain f.x1 ; x2 ; x3 /jx3 > 0g endowed with the metric ds2 D .dx21 C dx22 C dx23 /=.x3 /2 , is given by  D x32 .@2x1 C @2x2 C @2x3 /  x3 @x3 : The heat kernel of  is obtained from formula (3.7.32) by letting scalar curvature R D 1 and hyperbolic distance d.x0 ; x/ D , KD

2 1  e t e  4t : 3=2 .4 t/ sinh 

This relation was obtained by a direct method in [39], p. 396. The three-dimensional unit sphere S 3 . The Laplace–Beltrami operator on S 3 in spherical coordinates is given by D

1 @2 @ C C cot  @ 2 @ sin2 



@2 @2 @2 C  2 cos  2 2 @ @ @@

:

In this case R D 1 and the heat kernel becomes K.x0 ; xI t/ D

 t  2 1 e e 4t ; 3=2 .4 t/ sin 

(3.7.33)

where  D dS 3 .x0 ; x/. This was obtained for S U.2/ by Schulman [101], who also conjectured that this formula works in general for Lie groups. The three-dimensional Euclidean space. In the case of the Laplace operator D

@2 @2 @2 C C ; @x12 @x22 @x32

making the curvature R D 0 in (3.7.32) yields the familiar formula for the Euclidean heat kernel jxx0 j2 1  4t K.x0 ; xI t/ D e : .4 t/3=2

3.8 Heat Kernel at the Cut-Locus The point x belongs to the cut-locus of x0 if there is more than one geodesic between the points x0 and x in time t, and this number is finite. In this case the heat propagates from x0 to x in more than one way, each geodesic having its own contribution

3.10 Heat Kernel on the Half-Line

47

toward the heat kernel. If there are k geodesics between x0 and x, parameterized by Œ0; t, with the corresponding volume element Vk and classical action Sk , then the formula for the heat kernel is the sum of all contributions: K.x0 ; xI t/ D

k X

Vk .t; x0 ; x/e Sk :

(3.8.34)

j D1

The above sum has only one term in the case of elliptic operators. In the case of sub-elliptic operators the sum may become an infinite series, as in the case of the Grushin operator.

3.9 Heat Kernel at the Conjugate Locus If the points x0 and x are conjugate, there is a smooth variation with geodesics of the same length with fixed endpoints x0 and x. Consider the space x0 ;xIt D fc W Œ0; t ! M I c.0/ D x0 ; c.t/ D x; c geodesicg: Each element of the above space contributes to the propagation of heat between x0 and x. In this case the sum (3.8.34) becomes an integral over x0 ;xIt : Z

e S./ d ./;

K.x0 ; xI t/ D x0 ;xIt

where d ./ is the volume measure obtained by a limit process from Vk . This recovers a refined concept of path integral. In the particular case of homogeneous spaces, the volume element depends only on the Riemannian distance between x0 and x and does not depend in an explicit way on the end points. This can be seen in the cases of the Euclidean space and the three-dimensional hyperbolic space.

3.10 Heat Kernel on the Half-Line We interested in finding the heat kernel of the Laplacian

1 d2 2 dx2

on the domain Œ0; 1/,

D 0. This means that no heat can leak subject to the boundary condition across the point x D 0. Let x0 ; x > 0. The heat can travel from the point x0 to the point x in time t in two different ways: @ u.t; 0/ @x

 On the shortest path between the points, of length d1 D jx  x0 j, given by x.s/ D x0 C

.x  x0 /s ; t

s 2 Œ0; t:

48

3 The Geometric Method

 On a piecewise path that hits the origin at time and then is reflected toward x:  .1  s /x0 ; 0  s  ; x.s/ D x.s  /=.t  /;  s  t: of length d2 D x0 C x. The heat kernel is the amount of heat received by x at time t, which is the sum of the aforementioned amounts of heat: K.x0 ; xI t/ D p D p

d2 1

1

e  2t C p

2 t 1

e

2 t

.xx0 /2  2t

d2 2

1 2 t

e  2t

Cp

1 2 t

e

.xCx0 /2 2t

:

The following result will be needed in the next two applications. p Lemma 3.10.1. For any  2 R, T > 0, i D 1, we have X

1

2

e  2t .2nT C/ C

n1

X

1

2

2

e  2t .2nT / D 2e  2t

n1

X

e

2n2 T 2 t

 cos

n1

2nT  i : t

Proof. Expanding the binomial in the exponent and using Euler’s formula, we have X

1

2

e  2t .2nT C/ C

n1

D

X

X n1

e

2  2t

X

e

2 2  4n2tT



2  2t

X

e

2 2  2n t T

C

X

1

e

2nT  t

 cos

n1

Ce

2 T 2 C 2 4nT /

e  2t .4n

n1

n1

D 2e

2

1  2t .4n2 T 2 C 2 C4nT /

n1

De

1

e  2t .2nT /

2nT  t





2

D 2e  2t

X

e

2n2 T 2 t

cosh

n1

2nT  t

2nT  i : t 

3.11 Heat Kernel on S 1 Let X0 and X be two points on the unit circle S 1 with angular arguments s0 and s. The heat kernel K.X0 ; X I t/ depends on all geodesics that join the points X0 and X . The square of the lengths of both clockwise and counterclockwise geodesics joining X0 and X are `20 D .ss0 /2 ;

 2 `2n D 2nC.ss0 / ;

 2 `2n D 2n.ss0 / ;

n 2 N:

3.12 Heat Kernel on the Segment Œ0; T 

49

Since the unitpcircle is locally Euclidean and one-dimensional, the volume element is V .t/ D 1= 2 t, and the heat kernel has the following expression: K.X0 ; X I t/ D 0 1

@e Dp 2 t

X

`2 k

V .t/e  2t D p

n2Z .ss /2  2t0

C

X

e

1  2t

1 X

1 2 t 

1

e  2t



2

2nC.ss0 /

nD1

2nC.ss0 /

2

C

n1

X

e

1  2t



2n.ss0 /

2

1 A:

n1

Applying Lemma 3.10.1 with T D  and  D s  s0 , we obtain 0 1 K.X0 ; X I t/ D p e 2 t

.ss /2  2t0

@1 C 2

X

e

2 2  2 t n

n1

1 2n .s  s0 /i A : cos t 

Choosing z D .s  s0 /i=t and D 2=t in the definition of the theta-function 3 .zji / D 1 C 2

X

e  n

2

cos.2nz/

n1

[see (1.5.4)], we arrive at the following formula for the heat kernel on S 1 : K.X0 ; X I t/ D p

1 2 t

e

.ss0 /2 2t

 3

.s  s0 /i ˇˇ 2 i : ˇ t t

(3.11.35)

Since there is no generating formula for 3 , this shows that even in the simplest compact case, the heat kernel does not have a neat formula. For an approach using eigenfunctions expansion and the Poisson summation formula, see Sect. 6.4.

3.12 Heat Kernel on the Segment Œ0; T  The heat kernel of concern here is the one corresponding to the Neumann boundary @ @ conditions @x u.t; 0/ D @x u.t; T / D 0. This corresponds to the case when x D 0 and x D T are perfectly insulated reflecting walls. The distances traveled by heat from x0 to x in a direct way or using reflections in the walls are x  x0 ;

x C x0 ;

2nT C x  x0 ;

2nT  x C x0 ;

2nT C x C x0 ;

2nT  x  x0 ; : : : ;

50

3 The Geometric Method

with n D 1; 2; : : : . These are the distances from x0 to the images of x seen in two parallel plane mirrors situated at coordinates 0 and T . Each of these distances has a contribution to the heat kernel as follows: X `2 n V .t/e  2t K.x0 ; xI t/ D n0

D p

2

1 2 t

4e 

.xx0 /2 2t

C

X

C e

.xCx0 /2 2t

1

2

e  2t .2nT Cxx0 / C

n1

C

X

X

1

e  2t .2nT xCx0 /

2

n1

e

1  2t

.2nT CxCx0

n1

/2

C

X

3 e

1  2t

.2nT xx0

/2

5:

n1

Lemma 3.10.1 enables us to write the previous expression as 3 2  X 2n2 T 2 .xx0 /2 1 2nT K.x0 ; xI t/ D p e  2t 41 C 2 e  t cos .x  x0 /i 5 t 2 t n1 3 2  X 2n2 T 2 .xCx0 /2 2nT 1 e  2t 41 C 2 e  t cos .x C x0 /i 5 ; Cp t 2 t n1 which, after using the definition of the 3 -function, can be expressed as  ˇ 2T 2 i .xx0 /2 1 T ˇ e  2t 3 K.x0 ; xI t/ D p (3.12.36) .x  x0 /i ˇ t t 2 t  ˇ 2T 2 i .xCx0 /2 T 1 ˇ : (3.12.37) .x C x0 /i ˇ Cp e  2t 3 t t 2 t

3.13 Heat Kernel on the Cylinder The cylinder is two-dimensional and locally Euclidean, so the volume element is V .t/ D 21 t . Let .x0 ; y0 / and .x; y/ be two points on the cylinder. The geodesic joining the points is a spiral which makes a constant angle  with the vertical. If the geodesic winds n times, the vertical distance is the sum of n equal increments plus a remainder amount (see Fig. 3.2): y  y0 D 2n tan  C .x  x0 / tan : This shows that the angle  D n must be quantized by the relation tan n D

y  y0 ; 2n C .x  x0 /

n 2 Z:

3.13 Heat Kernel on the Cylinder

51

Fig. 3.2 The geodesic between the points .x0 ; y0 / and .x; y/ on the cylinder R  S1 tan

(x0 , y0)

x0

x0 mod

By convention, we consider the integer n positive if the winding is counterclockwise and negative otherwise. The length of the geodesic is the sum of n C 1 oblique segments: 2 n x  x0 `n D C : cos n cos n Expressing cosine in terms of tangent function and using the aforementioned formula yields q p 2  `n D D 1 C tan2 n .2n C x  x0 / D 2n C .x  x0 / C .y  y0 /2 : Then the heat kernel is given by the contribution of all geodesics from .x0 ; y0 / and .x; y/:   K .x0 ; y0 /; .x; y/I t   1 X  2t1 .2nCxx0 /2 C.yy0 /2 1 X  `2n e 2t D e D 2 t n2Z 2 t n2Z   1  .xx0 /2 C.yy0 /2 X  2t1 4n2  2 C4n.xx0 / 2t e D e 2 t n2Z  2n.x  x0 /i 1  .xx0 /2 C.yy0 /2 X  2n2  2 2t t e cos e D 2 t t n2Z  2n.x  x0 /i i  .xx0 /2 C.yy0 /2 X  2n2  2 2t t e sin e C 2 t t n2Z 1 0  2 C.yy /2 X 2 2 .xx / 2n.x  x0 /i A 1  2n  0 0 @1 C 2 2t e  t cos e D 2 t t n1 1  .xx0 /2 C.yy0 /2 2t e 3 D 2 t



.x  x0 /i ˇˇ 2 i : ˇ t t

52

3 The Geometric Method

It is worth noting that the previous heat kernel is the direct product of the heat kernels on R and S 1 . This formula could have been written directly since the cylinder is the product of these two manifolds.

3.14 Operators with Potentials This is an extension of the method described in the previous sections to operators that have a potential. We start the presentation with the heat kernel of the operator 1 d2 C U.x/, where U.x/ is a function of the variable x, called the potential. Since 2 dx the heat kernel will depend on the classical action between any two given points, in order to obtain closed-form formulas, we need to consider only those potentials U.x/ for which the classical action can be computed explicitly. This will occur in several cases, for instance, in the case of linear and quadratic potentials. In these cases there is a unique solution joining any two given points x0 and x within time t. The classical action cannot be obtained explicitly for potentials of the type U.x/ D a2 x m , with m  3. d 2 Consider the operator L D 12 dx C U.x/ with smooth potential U.x/. We associate the Hamiltonian function H.p; x/ D

1 2 p C U.x/: 2

(3.14.38)

Hamilton’s equations are xP D Hp D p; pP D Hx D U 0 .x/; and hence xR D pP D U 0 .x/: For any two given points x0 and x, the classical path joining them is obtained by solving the equation 8 0 ˆ ˆ <xR D U .x/; (3.14.39) x.0/ D x0 ; ˆ ˆ :x.t/ D x: We shall assume the potential U.x/ is such that the previous system has a unique solution. Since the Hamiltonian (3.14.38) does not depend explicitly on the variable t, it will be preserved along the solutions of (3.14.39): H D

xP 2 C U.x/ D E; 2

(3.14.40)

where E D E.x0 ; x; t/ is the constant of energy. Hence x.s/ verifies the integral equation Z x.s/ dw D ˙s; p 2E  2U.w/ x0

3.14 Operators with Potentials

53

where the positive (negative) sign is taken on the right-hand side if x > x0 .x < x0 /. The energy E D E.x0 ; x; t/ satisfies the equation Z x dw p D ˙t; 2E  2U.w/ x0 with the same sign convention. The action S is verifying the Hamilton–Jacobi equation @t S D E.x0 ; x; t/. Since along the solutions we have p D @x S , using xP D p, we get xP D @x S , and hence (3.14.40) becomes .@x S /2 D 2E  2U.x/:

(3.14.41)

We shall look for a fundamental solution of the type K D V .t; x/e S , where S D Scl is the classical action. A computation provides  0 V CE ; @t K D K V  @x V C @x S ; @x K D K V  2 @ @x V xV 2 @x S C .@x S /2  @2x S : @2x K D K V V Let P D @t  12 @2x  U.x/. Using the above formulas, we obtain  PK D K DK

1 @t V 1 @2x V @x V @x S 1 CE  C  .@x S /2 C @2x S  U.x/ V 2 V V 2 2 ! 1 2 @t V  2 @x V C @x V @x S 1 C @2x S ; V 2



where we used (3.14.41). Hence PK D 0 if and only if V satisfies the generalized transport equation 1 1 @t V  @2x V C @x S @x V C @2x S V D 0: 2 2

(3.14.42)

Using that xP D @x S , we have d P x V D @t V C @x S @x V; V .t; x/ D @t V C x@ dt and hence (3.14.42) becomes d 1 1 V .t; x/ D @2x V .t; x/  @2x S V .t; x/: dt 2 2

(3.14.43)

54

3 The Geometric Method

If @2x S depends on t only, then it makes sense to look for a function V D V .t/ satisfying the simplified transport equation 1 @t V C @2x S V D 0: 2

(3.14.44)

The volume function V .t; x/ should satisfy the initial condition Z lim V .t; x/e Scl dx D 1: t &0

(3.14.45)

We conclude the above calculations with the following method for finding explicit formulas for heat kernels: 1. Assume the boundary value problem (3.14.39) satisfies the uniqueness condition; let x.s/ be its solution. 2. Let S D Scl .x0 ; xI t/ be the classical action along solution x.s/ between x0 and x in time t. 3. Consider the solution V .t; x/ of the transport equation (3.14.42) satisfying (3.14.45). 4. Then K.x0 ; xI t/ D V .t; x/e S is the heat kernel for the operator 12 @2x C U.x/. Unfortunately, there are potential functions U.x/ for which the boundary value problem (3.14.39) has a unique solution. This happens, for instance, in the case when U.x/ D x 4 ; see Calin and Chang [24]. Even if the solution is unique and the action function is found, the transport equation might be difficult to solve. It is worth noting that the potential function U.x/ does not appear explicitly in the expression of the generalized transport equation. It is even more interesting to observe that (3.14.42) is a particular form of the multidimensional generalized transport equation (3.4.11) with .aij / D 1 (one-dimensional matrix). In the following se shall consider the case of linear and quadratic potentials, for which all the computations can be done explicitly.

3.15 The Linear Potential Let U.x/ D ax, with a a real constant. The operator is L D associated Hamiltonian function is H.p; x/ D

1 2 p  ax: 2

The Hamiltonian system becomes xP D Hp D p; pP D Hx D a:

1 d2 2 dx

 ax and its

3.15 The Linear Potential

55

The associated Lagrangian function is L.x; x/ P D 12 xP 2 C ax. The Euler–Lagrange equation xR D a with the boundary conditions x.0/ D x0 , x.t/ D x has the unique solution 1 x.s/ D as 2 C bs C x0 ; (3.15.46) 2 with x  x0 at  : (3.15.47) bD t 2 The Lagrangian along the solution (7.9.43) is   1 P 2 C ax.s/ L x.s/; x.s/ P D x.s/ 2  1 1 2 D .as C b/2 C a as C bs C x0 2 2 1 D a2 s 2 C 2abs C ax0 C b 2 : 2 The classical action is obtained by integrating the Lagrangian along the solution: Z

t

S.x0 ; xI t/ D

  L x.s/; x.s/ P ds

0

1 1 D a2 t 3 C abt 2 C ax0 t C b 2 t: 3 2 Substituting b from (7.10.67) yields   1 2 3 1 x  x0 at 2 at 2 x  x0 S.x0 ; xI t/ D a t C at C ax0 t C t   3 t 2 2 t 2 .x  x0 /2 1 2 3 1 D (3.15.48) C a.x C x0 /t  a t : 2t 2 24 There is an alternate method for finding the action using the Hamilton–Jacobi equation. For this we need first to find the energy along the solution as a function of the boundary points. In our case (3.14.40) becomes 1 x.s/ P 2  ax.s/ D E: 2 Using (7.9.43) yields ED

 1 2 1 .as C b/2  a as C bs C x0 ; 2 2

8s 2 Œ0; t:

56

3 The Geometric Method

Taking the particular value s D 0 yields 1 2 b  ax0 2  1 x  x0 at 2 D  ax0  2 t 2

ED

D

.x  x0 /2 1 a2 t 2 a.x C x :  / C 0 2t 2 2 8

It is not hard to check that the Hamilton–Jacobi equation @t S D E 1 a2 t 2 .x  x0 /2 C /  a.x C x ; D 0 2t 2 2 8 with the initial condition Sjt D0 D 0, has the solution S.x0 ; xI t/ D

1 .x  x0 /2 1 C a.x C x0 /t  a2 t 3 : 2t 2 24

(3.15.49)

In the following we shall find the volume function V . Since 12 @2x S D V D V .t/ and the transport equation is V 0 .t/ C

1 V .t/ D 0; 2t

with the solution V .t/ D

c t 1=2

:

The constant c can be found using (3.14.45): Z Z 1 D lim K.x0 ; xI t/ dx D lim V .t/ e S.x0 ;xIt / dx t &0

D lim

c

D lim

c

e

t &0 t 1=2

t &0

D lim

t

t &0

Z

.xx0 /2 1 2 3  1 2t 2 a.xCx0 /t C 24 a t

1

1

2t3

1

1

2t3

e  2 ax0 t C 24 a 1=2 c

t &0 t 1=2

e  2 ax0 t C 24 a

Z

e

.xx0 /2 2t

1

dx

1 2 a.xx0 /t

.2 t/1=2 e 8 a

dx

2t 3

D c.2/1=2 ; and hence c D .2/1=2 and the volume function is V .t/ D 1=.2 t/1=2 .

1 , 2t

then

3.15 The Linear Potential

57

We have arrived at the following result: Theorem 3.15.1. Let a 2 R. The heat kernel of the operator L D given by K.x; x0 ; t/ D p

1 2 t

e

.xx0 /2 2t

1 2 3 1 2 a.xCx0 /t C 24 a t

1 d2 2 dx

 ax is

; t > 0:

Remark 3.15.2. In the case when a D 0 we recover the well-known heat kernel of 1 2 @ , 2 x

which is

p1 2 t

e

.xx0 /2 2t

.

Without muchPeffort, the previous result can be generalized to the n-dimensional P operator L D 12 niD1 @2xi  niD1 ai xi . In this case the Hamiltonian function is H.p; x/ D

1X 2 X pi  ai xi ; 2 n

n

i D1

i D1

and the Hamiltonian equations are xP j D pj ; pPj D aj ;

j D 1; : : : ; n:

The solution joining the points x0 and x is x j .s/ D

1 j aj s 2 C bj s C x0 ; 2

(3.15.50)

with bj D

x j  x0j aj t  ; t 2

j D 1; : : : ; n:

(3.15.51)

The Lagrangian L.x; x/ P D

1X 2 X 1 jxP i j2 C ha; xi D xP i C ai xi 2 2 n

n

i D1

i D1

evaluated along the previous solution is   1 L x.s/; x.s/ P D jaj2 s 2 C 2ha; bis C ha; x0 i C jbj2 : 2 Integrating the Lagrangian along the solution yields the following classical action: S.x0 ; xI t/ D

1 jx  x0 j2 1 C ha; x C x0 it  jaj2 t 3 : 2t 2 24

(3.15.52)

58

3 The Geometric Method

Similar calculations show that the generalized transport equation in this case is ! n n 1X 2 1 X 2 @t V  @xi V C hrS; rV i C @xi S V D 0: 2 2 i D1 i D1 P n does not depend on x, it makes sense to look for a Since 12 niD1 @2xi S D 2t volume function of the form V D V .t/, in which case the previous equation takes the simplified form n V 0 .t/ C V .t/ D 0: 2t The solution is V .t/ D ct n=2 . Using a similar method as before, we find c D .2/n=2 . We conclude with the following result: P Theorem 3.15.3. Let a 2 Rn . The heat kernel of the operator L D 12 niD1 @2xi  Pn i D1 ai xi is given by K.x; x0 ; t/ D

jxx0 j2 1 1 2 3  1 2t 2 ha; xCx0 it C 24 jaj t ; t > 0: e .2 t/n=2

3.16 The Quadratic Potential 2

d We consider the operator L D 12 . dx  a2 x 2 /, with a constant, called the Hermite operator. The associated Hamiltonian is given by H.p; x/ D 12 p 2  12 a2 x 2 and the Hamiltonian system of equations is

xP D Hp D p; pP D Hx D a2 x: The geodesic between x0 and x within time t satisfies xR D a2 x; x.0/ D x0 ; x.t/ D x:

(3.16.53)

x  x0 cosh.at/ sinh.as/ C x0 cosh.as/: sinh.at/

(3.16.54)

The solution is x.s/ D

The Lagrangian associated with the previous Hamiltonian is L.x; x/ P D p xP  H D

1 2 1 2 2 xP C a x : 2 2

(3.16.55)

3.16 The Quadratic Potential

59

Integrating the solution (7.9.51) along the Lagrangian (7.9.52) and performing the same computations as in Sect. 7.8 yields S.x0 ; x; t/ D

 2  a .x C x02 / cosh.at/  2xx0 : 2 sinh.at/

(3.16.56)

There is an alternate way of finding the action using the Hamilton–Jacobi equation. The conservation of energy law for (7.9.50) is 1 2 1 xP .s/  a2 x 2 .s/ D E; 2 2 where E is the energy constant. This can also be written as p dx dx D 2E C a2 x 2 H) p D ds: ds 2E C a2 x 2 Integrating between s D 0 and s D t, with x.0/ D x0 and x.t/ D x, and solving for the energy yields   a2 x 2 C x02  2xx0 cosh.at/ a2 .x  x0 cosh.at//2 2 2 2E D  a x0 D : sinh.at/2 sinh.at/2 The Hamilton–Jacobi equation becomes @t S D H.rS / D E   a2 x 2 C x02  2xx0 cosh.at/ D 2 sinh.at/2

 @ a 2 axx0 2 D : .x C x0 / coth.at/  @t 2 sinh.at/ Hence

 a 2xx0 2 2 .x C x0 / coth.at/  S.x0 ; x; t/ D 2 sinh.at/ D

 2 

1 a x C x02 cosh.at/  2xx0 : 2 sinh.at/

(3.16.57)

Since @2x S D a coth.at/ is a function of t only, the transport equation becomes 1 V 0 .t/ C a coth.at/V .t/ D 0: 2

60

3 The Geometric Method

Integrating, we obtain the solution c V .t/ D p ; sinh.at/ where c is a constant which will be determined later. Using K D V .t/e S , we get the fundamental solution K.x0 ; x; t/ D p

c sinh.at/

a

1

e  2 sinh.at / Œ.x

2 Cx 2 / cosh.at /2xx  0 0

:

In order to determine the constant c we write s 1 at 2 2 c at e  2t  sinh.at / Œ.x Cx0 / cosh.at /2xx0  : K.x0 ; x; t/ D p at sinh.at/ Since at= sinh.at/ ! 1, for a ! 0, we can write 1 c 2 K.x0 ; x; t/  p e 2t .xx0 / : at

By comparison with the fundamental p solution for the usual heat operator p 1 2 .xx / 0 , we obtain c D a=2. To conclude, we have the following 1= 2 t e 2t result. 2

d Theorem 3.16.1. Let a be a constant. The heat kernel for the operator 12 . dx  2 2 a x / is s 1 at 2 2 1 at e  2t sinh.at / Œ.x Cx0 / cosh.at /2xx0  ; t > 0: K.x0 ; x; t/ D p 2 t sinh.at/

Using the substitutions a D i ˛ and cosh.i ˛t/ D cos.˛t/ and sinh.2i ˛t/ D i sin.2˛t/, we get 2

d Corollary 3.16.2. Let ˛ be a constant. The heat kernel for the operator 12 . dx C 2 2 ˛ x / is s 1 2˛t 2 2˛t 2 1 K.x0 ; x; t/ D p e  2t sin.2˛t / Œ.x Cx0 / cos.2˛t /2xx0  ; t > 0: 2 t sin.2˛t/

3.17 The Operator

1 2

P

@x2 i ˙ 12 a2 jxj2

Consider the operator 1 1 1 1 n  a2 jxj2 D .@2x1 C    C @2xn /  a2 .x12 C    C xn2 /; a  0: 2 2 2 2

3.17 The Operator

1 2

P

@2xi ˙ 12 a2 jxj2

61

The associated Hamiltonian is H D

1 2 1 .1 C    C n2 /  a2 .x12 C    C xn2 /; 2 2

with the Hamiltonian system ( xP j D Hj D j ; Pj D Hxj D a2 xj ; j D 1; : : : ; n: The geodesic x.s/ starting at x0 D .x10 ; : : : ; xn0 / and having the final point x D .x1 ; : : : ; xn / satisfies the equations 8 2 ˆ ˆ <xR j D a xj ; xj .0/ D xj0 ; ˆ ˆ :x .t/ D x ; j D 1; : : : ; n: j

j

As in the one-dimensional case, we have the law of conservation of energy xP j2 .s/  a2 xj2 .s/ D 2Ej ; j D 1; : : : ; n; where Ej is the energy constant for the j th component. The total energy, which is the Hamiltonian, is given by H D

n  X 1

1 xP j2  a2 xj2 2 2

j D1

D E1 C    C En D E(constant):

Since the energy for the one-dimensional case is Ej D

a2 Œxj2 C .xj0 /2  2xj xj0 cosh.at/ 2 sinh2 .at/

;

we get the total energy H DED

n X j D1

where jxj2 D

Pn

j D1

Ej D

a2 Œjxj2 C jx0 j2  2hx; x0 i cosh.at/ ; 2 sinh2 .at/

xj2 and hx; x0 i D

Pn

0 j D1 xj xj .

Computing the classical action. The action between x0 and x within time t satisfies the equation @t@ S D E or @ a2 Œjxj2 C jx0 j2  2hx; x0 i cosh.at/ S D @t 2 sinh2 .at/

 ahx; x0 i @ a 2 2 .jxj C jx0 j / coth.at/  D @t 2 sinh.at/

62

3 The Geometric Method

and hence Scl .x0 ; xI t/ D

a 1 .jxj2 C jx0 j2 / cosh.at/  2hx; x0 i : 2 sinh.at/

(3.17.58)

We are looking for a kernel of the form K.x0 ; x; t/ D V .t/e kScl .x0 ;x;t / ;

k 2 R:

(3.17.59)

Since the Lagrangian is at most quadratically in x and x, P the function V .t/ is given by the van Vleck formula (see also Chap. 7) s  1 @2 Scl V .t/ D det  : 2 @x @x0 Since in this case a @2 Scl D In ; @x @x0 sinh.at/ we obtain

 V .t/ D

a 2 sinh.at/

n=2 :

Hence the heat kernel is  K.x0 ; x; t/ D

a 2 sinh.at/

n=2

a

1

e  2 sinh.at /



.jxj2 Cjx0 j2 / cosh.at /2hx;x0 i

for t > 0. In a similar way, one can find the heat kernel for the operator 12 n C 12 a2 jxj2 . Formally, this reduces to changing sinh into sin and cosh into cos in the previous expression. An alternate method. Another method for finding the heat kernel of 1X 2 @xi  2ajxj2 2 n

LD

i D1

is described next by following reference [11]. Using the rotational symmetry of the operator L, it makes sense to state the following. Anzatz: The heat kernel of L starting at the origin is of the form 1

2

K.x; aI t/ D a .t/e  2 ˛a .t /jxj ;

t > 0:

3.17 The Operator

1 2

P

@2xi ˙ 12 a2 jxj2

63

Substituting in equation .@t  L/K D 0 yields  0  2 1 2  .t/ n 0 2 ˛ .t/ C ˛ .t/  a jxj K D 0:  ˛.t/ C  .t/ 2 2 We shall look for functions  and ˛ such that ˛ 0 .t/ D a2  ˛ 2 .t/; n  0 .t/ D  ˛.t/: .t/ 2 Using the substitution ˛ D ˇ 0 =ˇ yields the equation ˇ 00 D a2 ˇ, with the solution ˇ.t/ D A cosh.at/ C B sinh.at/: Using limt &0C K.x; aI t/ D ıx yields 1 D lim ˛.t/ D t &0C

ˇ 0 .0/ 2aB D ; ˇ.0/ A

so A D 0. Hence ˛.t/ D a coth.at/. Integrating in

 0 .t / .t /

D  n2 a coth.at/ yields .t/ D

cn : sinh.at/n=2

The constant cn may be obtained from the condition Z

lim

t &0C

1 cn 2 e  2 ˛a .t /jxj dx D 1 ” sinh.at/n=2  cn 2 n=2 lim D1” ˛.t/ t &0C sinh.at/n=2 n=2  2 lim D cn1 ” t &0C a cosh.at/  a n=2 : cn D 2

Hence we obtain the heat kernel of L: 1

2

K.x; aI t/ D a .t/e  2 ˛a .t /jxj n=2  a 1 2 D e  2 a coth.at /jxj ; 2 sinh.at/

t > 0:

64

3 The Geometric Method

3.18 The Operator L D 12 @x2 C

 x2

The operator is defined on the domain fx > 0g. This is an example where the transport equation reduces to a modified Bessel equation. This operator has been considered from the geometric point of view in reference [26]. First we find the classical action. The associated Hamiltonian is H D 12 p 2 C x 2 , a real constant. The geodesic joining the points x0 and x within time t satisfies the Euler–Lagrange equation associated with the Lagrangian L D 12 xP 2  x 2 : 2 ; x3 x.0/ D x0 ; x.t/ D x: xR D

Since the regions fx < 0g and fx > 0g are separated, in order to have connectivity, we have to assume that either x0 ; x > 0 or x0 ; x < 0. We can show that the energy is a first integral of motion, so that 1 2 xP .s/ C 2 D E; 2 x .s/

s 2 Œ0; t:

Under the assumption x0 ; x > 0, the previous relation becomes x.s/x.s/ P D

p

2Ex 2 .s/  2 :

Let u.s/ D x 2 .s/, u0 D x02 , ut D x 2 . Then u.s/ verifies the ODE p uP D 2 2Eu  2 ; u.0/ D u0 ;

u.t/ D ut :

Integrating yields Z

ut

du p D 2t ” 2Eu  2 u0 p p 2Eut  2  2Eu0  2 D 2Et: Eliminating the square roots, we obtain  2 .u0 C ut /  2Et 2 D 4.u0 ut  2 t 2 /;

(3.18.60)

3.18 The Operator L D 12 @2x C

x2

65

assuming the following condition:
0;

(8.13.62)

is given by pt . 0 ; / D

 1 .2 C2 / 

0  I0 ; e 2t 0 2 t t

t > 0; > 0:

(8.13.63)

Since the heat kernel of a sum of commuting operators is the product of their kernels, we obtain the following consequence.

8.14 Brownian Motion on a Circle

181

Corollary 8.13.4. The heat kernel of the operator n 1 X @2 1X 1 @ C 2 2 xi @xi @xi2 i D1n i D1

is K.x0 ; xI t/ D

n xi xi0 x1    xn  1 .jx0 j2 Cjxj2 / Y 2t ; e I 0 .2 t/n t

xi ; xi0 ; t > 0:

i D1

8.14 Brownian Motion on a Circle Let x0 D .cos 0 ; sin 0 / 2 S 1 be a fixed point on the unit circle. If Wt is the one-dimensional Brownian motion, consider the following system of stochastic differential equations: 1 dX 1 .t/ D  X1 .t/ dt  X2 .t/ dW t ; 2 1 dX 2 .t/ D  X2 .t/ dt C X1 .t/ dW t ; 2

(8.14.64) (8.14.65)

to Theorem 8.2.2, the aforementioned with the initial condition X0 D x0 . According   system has a unique solution X.t/ D X1 .t/; X2 .t/ . If we let     Xt D X1 .t/; X2 .t/ D cos.0 C Wt /; sin.0 C Wt / ;

(8.14.66)

a direct application of Ito’s formula yields 1 dX 1 .t/ D  cos.0 C Wt / dt  sin.0 C Wt / dW t ; 2 1 dX2 .t/ D  sin.0 C Wt / dt C cos.0 C Wt / dW t ; 2 which shows that (8.14.66) is the desired solution for the system (8.14.64)– (8.14.65). The process (8.14.66) is called the Brownian motion on the circle S 1 starting at x0 . The stochastic differential equation can also be written as dX t D b.Xt / dt C .Xt / dW t ; with the drift and diffusion given by

x1 =2 ; b.x/ D x2 =2

x2 .x1 ; x2 / D : x1

182

8 The Stochastic Analysis Method

Then  T D



x1 x2 ; x12

x22 x1 x2

and hence the generator of the process Xt is AD

X @ 1X @2 . T /ij C bj .x/ 2 @xi @xj @xj i;j

j

D

i 1h 2 2 x2 @x1 C x12 @2x2  2x1 x2 @2x1 x2  x1 @x1  x2 @x2 2

D

1 .x1 @x2  x2 @x1 /2 : 2

In polar coordinates x1 D r cos , x2 D r sin , we have A D 12 @2 , which is the Laplace operator on S 1 . The heat kernel of A is given by the transition density of the process   Xt D cos.0 C Wt /; sin.0 C Wt / : The distribution function given that the Brownian motion on the circle starts from 0 at t D 0 is F . j 0 / D P .0 C Wt C 2n < ; 8n 2 Z/ X X D P .Wt  2n C   0 / D FWt .2n C   0 /; n2Z

n2Z

and then the transition density is pt .0 ; / D

X 1 .2nC0 /2 d 2t e F .j0 / D p d 2 t n2Z

D p D p

1

X

2 t

n2Z

1 2 t

e

1

e  2t



.0 /2 2t

4n2  2 C.0 /2 C4n.0 /

X

e

2n2  2 t

D p

2 t

2n.0 / t

n2Z

0 1

e

e

.0 /2  2t

@1 C 2

X n1

e



2 2  2n t 

1 2n.  0 / A cosh : t

This series can also be expressed as a theta-function, see (3.11.35).

8.15 An Example of a Heat Kernel

183

8.15 An Example of a Heat Kernel The following example follows [25]. Consider the stochastic differential equation dX t D

q

1C

Xt2

1 C Xt 2

dt C

q

1 C Xt2 dW t ;

X0 D x0 :

The drift and diffusion coefficients are given by b.x/ D p 1 C x 2 . The following estimations hold: p p j.x/  .y/j D j 1 C x 2  1 C y 2 j D p D p

p

(8.15.67)

1 C x 2 C 12 x, .x/ D

jx 2  y 2 j p 1 C x2 C 1 C y 2

jx C yj  jx  yj < 2jx  yj; p 1 C x2 C 1 C y 2

since jx C yj jxj jyj p p p  p Cp p 2 2 2 2 2 1Cx C 1Cy 1Cx C 1Cy 1 C x C 1 C y2 < p

jxj

Cp

jyj

< 1 C 1; 1 C x2 1 C x2 p p 1 jb.x/  b.y/j  j 1 C x 2  1 C y 2 j C jx  yj 2 5 1 < 2jx  yj C jx  yj D jx  yj: 2 2 Hence jb.x/  b.y/j C j.x/  .y/j 

9 jx  yj; 2

8x; y 2 R;

and by Theorem 8.2.2 it follows that the stochastic equation (8.15.67) has a unique solution. Therefore, instead of solving the equation, it suffices to guess a solution. Let Ut D c Ct CWt , where c D sinh1 .x0 /. Then Xt D sinh.Ut /, with X0 D x0 , and by Ito’s formula we have 1 sinh Ut .d Ut /2 2 1 D cosh.Ut / .dt C dW t / C sinh.Ut / dt 2   1 D cosh.Ut / C sinh.Ut / dt C cosh.Ut / dW t 2

dX t D cosh.Ut / d Ut C

184

8 The Stochastic Analysis Method

q



q 1 1 C sinh2 .Ut / C sinh.Ut / dt C 1 C sinh2 .Ut / dW t 2

q q 1 D 1 C Xt2 C Xt dt C 1 C Xt2 dW t : 2 D

Hence Xt D sinh.c C t C Wt /;

c D sinh1 .x0 /

(8.15.68)

is the solution of (8.15.67). The generator of the aforementioned process Xt is the operator p 1 d2 x d A D .1 C x 2 / 2 C 1 C x2 C : (8.15.69) 2 2 dx dx The conditional distribution function of Xt is FXjX0 .xjx0 / D P .Xt  x j X0 D x0 / D P .sinh.Ut /  x j U0 D sinh1 x0 / D P .c C t C Wt  sinh1 x j x D sinh1 x0 / D P .Wt  sinh1 x  c  t j c D sinh1 x0 / D P .Wt  sinh1 x  sinh1 x0  t/ Z sinh1 xsinh1 x0 t 1 u2 D e  2t du; p 2 t 1 so the transition density of Xt is pt .x0 ; x/ D

.sinh1 xsinh1 x0 t /2 d 1 2t ; e FXjX0 .xjx0 / D p dx 2 t

t > 0: (8.15.70)

We may further transform this formula by using

q   p sinh1 x  sinh1 x0 D ln x C 1 C x 2  ln x0 C 1 C x02 p x C 1 C x2 ; q D ln x0 C 1 C x02 p p x C 1 C x2 1 C x2 1 1 2 2 xC .sinh x  sinh x0  t/ D ln  2t ln C t 2: q q 2 2 x0 C 1 C x0 x0 C 1 C x0

8.16 A Two-Dimensional Case Fig. 8.7 The transition density as a function of x, with x0 D 0 in the cases: t D 1, t D 0:5 and t D 0:25. For t small, the graph tends to the Dirac distribution centered at x0 D 0

185

0.25

1

0.5

Then the transition density (8.15.70) becomes p p 1 t 2 xC 1Cx 2 1 C x 2  2t ln x0 Cp1Cx2  2 0 e ; pt .x0 ; x/ D p q 2 t x0 C 1 C x 2 0

1

xC

t > 0: (8.15.71)

To conclude, the heat kernel of the operator (8.15.69) is given by (8.15.71). Its graph is given by Fig. 8.7.

8.16 A Two-Dimensional Case Let W1 .t/ and W2 .t/ be two independent Brownian motions and consider the random process Xt D .X1 .t/; X2 .t// given by X1 .t/ D W1 .t/ C W2 .t/ C x10 ;   X2 .t/ D sinh W2 .t/ C c ; c D sinh1 .x20 /: The process starts at X0 D .x10 ; x20 / D x0 . The associated stochastic differential equation is dX 1 .t/ D dW 1 .t/ C dW 2 .t/;   1 dX 2 .t/ D cosh W2 .t/ C c dW 2 .t/ C X2 .t/ dt 2 p 1 D 1 C X2 .t/2 dW 2 .t/ C X2 .t/ dt: 2

186

8 The Stochastic Analysis Method

In matrix form this becomes







1p 1 0 dX1 .t/ dW 1 .t/ D 1 dt C ; X .t/ dX2 .t/ dW 2 .t/ 0 1 C X2 .t/2 2 2 with the coefficients b.x/ D

0 ; 1 x 2 2

D

0

0

!

1q 1

1 C x22

;

 T D @q

q 1

1 C x22

1 C x22 1 C x22

1 A:

Following relation (8.3.19), the generator of Xt is given by AD

X @ 1X @2 . T /ij C bj .x/ 2 @xi @xj @xj i;j

0

j

q

1

D

2 1 1 C x22 1@ A @ q C 2 @xi @xj 1 C x22 1 C x22

D

1 2





0  .@x1 ; @x2 / 1 x 2 2

q 2

@2 1 @2 2 @ 2 C 2 1 C x C .1 C x / C x2 @x2 : 2 2 2 2 @x1 @x2 2 @x1 @x2

In order to find the heat kernel of the operator A, we need to find the transition density of the process Xt D .X1 .t/;X2 .t//. This can  be obtained from the transition density of the Brownian motion W1 .t/; W2 .t/ using formula (8.6.23) and the technique presented in Sect. 8.6, item 2. The transformation x1 D u1 C u2 C x10 ; x2 D sinh.u2 C c/;

c D sinh1 .x20 /;

has the inverse   u1 D .x1  x10 /  sinh1 .x2 /  sinh1 .x20 / ;

(8.16.72)

u2 D sinh1 .x2 /  sinh1 .x20 /;

(8.16.73)

with the Jacobian det

@.u1 ; u2 / 1 6D 0: D q @.x1 ; x2 / 1 C x22

8.16 A Two-Dimensional Case

187

The density function of Xt is   @.u1 ; u2 / pt .x0 ; x/ D fX jX.0/Dx0 .xjx0 / D fW1 ;W2 u1 .x0 ; x/; u2 .x0 ; x/ det @.x1 ; x2 /     @.u1 ; u2 / D fW1 u1 .x0 ; x/ fW2 u2 .x0 ; x/ det @.x1 ; x2 / 2 u2 u 1 1 1 1 2 e  2t p e  2t q D p 2 t 2 t 1 C x22 D

juj2 1 1 q e  2t ; 2 t 1 C x 2 2

with u given by (8.16.72)-(8.16.73). A computation provides juj2 D u21 C u22

  D .x1 x10 /2  2.x1  x10 / sinh1 .x2 /  sinh1 .x20 / 2  C2 sinh1 .x2 /  sinh1 .x20 / q x2 C 1 C x22  2 D .x1 x10 /2 2.x1 x10 / ln C 2 sinh1 .x2 / sinh1 .x20 / q x20 C 1C.x20 /2

and hence

e

2  juj 2t

" D

0 q 1 # x1 x  2 t x2 C 1 C x22 1 0 2 1 1 1 0 q e  2t .x1 x1 /  t sinh .x2 /sinh .x2 / : x20 C 1 C .x20 /2

We have arrived at the following result. Proposition 8.16.1. The heat kernel of the operator 1 AD 2



q 2

@2 1 @2 2 @ 2 C 2 1 C x2 C .1 C x2 / 2 C x2 @x2 2 @x1 @x2 2 @x1 @x2

is given by 0 q 1 # x1 x 2 t C 1 C x x 2 2 1 1 q pt .x0 ; x/ D q 2 t 1 C x 2 x 0 C 1 C .x 0 /2 2 2 2  2 1 0 2 1 1 1 0 e  2t .x1 x1 /  t sinh .x2 /sinh .x2 / ; t > 0:

"

188

8 The Stochastic Analysis Method

8.17 Kolmogorov’s Operator One of the classical examples of heat kernels obtained by using the transition density of an associated stochastic process is the degenerate operator of Kolmogorov on R2 (see [80, 81]): 1 L D @2x1  x1 @x2 : (8.17.74) 2 Since





0 10 1 0 T ; bD  D ; ; D x1 00 0 0 the operator L is the generator of the following two-dimensional Ito diffusion: dX t D b dt C  dW





dW 1 .t/ 1 0 0 dt C ; D 0 0 X1 .t/ dW 2 .t/ where W1 .t/ and W2 .t/ are independent Brownian motions. If Xt D .X1 .t/; X2 .t//, then dX 1 .t/ D dW 1 .t/; dX 2 .t/ D X1 .t/ dt; with the solution X1 .t/ D x10 C W1 .t/; Z t Z t  0  X2 .t/ D x20  x1 C W1 .s/ ds D x20  x10 t  W1 .s/ ds; 0

0

where X0 D .x10 ; x20 /. We shall use the expected value method presented in item 6 of Sect. 8.6. Denote W .t/ D W1 .t/:     pt .x0 ; x/ D EŒı X1 .t/  x1 ı X2 .t/  x2    Z t D E ı.x1  x10  W .t//ı.x20  x2  x10 t  W .s/ ds/ 0   Z Rt 1 0 i.x20 x2 x10 t  0 W .s/ ds/ d e D E ı.x1  x1 C W .t// 2 Z Rt 1 0 0 e i .x2 x2 x1 t / EŒı.x10  x1 C W .t//e i 0 W .s/ ds  d: D 2 (8.17.75)

8.18 The Operator 12 x12 @2x2

189

Let v D v.x10 ; x1 ; tI / D EŒı.x10  x1 C W .t//e i

Rt 0

W .s/ ds

:

According to the Feynman–Kac formula (see Theorem 8.23.1) with the linear potential V .x/ D ax and a D i, the element v verifies the initial boundary problem @t v D

1 2

 @2x1  ax1 v;

vjt D0 D ıx 0 .x/: 1

Theorem 3.15.1 yields vD p

1 2 t

e

.x1 x 0 /2 1 2t

1 2 3 0 1 2 a.x1 Cx1 /t C 24 a t

.x1 x 0 /2 i 1 t3 2 0 1 e  2t  2 .x1 Cx1 /t  24  : D p 2 t

Substituting back in (8.17.75) yields .x1 x 0 /2 1 1 1 pt .x0 ; x/ D e  2t p 2 2 t

Z

t3

e  24 

2 CiŒx x 0  1 .x Cx 0 /t  2 1 2 2 1

d

r 0 /2 .x1 x1 1 24  243 Œx2 x20  12 .x1 Cx10 /t 2 1  2t e D e 4t p 2 2 t t3 p 0 2 1 /  6 Œx x 0  1 .x Cx 0 /t 2 3  .x1 x 2 1 2 2 1 2t t3 D e ; t > 0; t2

which is the heat kernel of the operator (8.17.74).

8.18 The Operator 12 x21 @x22 The operator A D 12 x12 @2x2 is the generator of the two-dimensional Ito diffusion dX 1 .t/ D 0; dX 2 .t/ D X1 .t/ dW t ;

t  0;

where Wt is a one-dimensional Brownian motion. Solving yields X1 .t/ D x10 ; X2 .t/ D x20 C x10 Wt :

190

8 The Stochastic Analysis Method

Assume x10 6D 0. Then the transition distribution function is   FXjX0 .x1 ; x2 j x10 ; x20 / D P X1 .t/  x1 ; X2 .t/  x2 j X1 .0/ D x10 ; X2 .0/ D x20 D P .x10  x1 ; x20 C x10 Wt  x2 /

x2  x20 0 D P .x1  x1 /P Wt  x10 Z D Hx 0 .x1 / 1

x2 x 0 2 x0 1

1

where

 Hx0 .x1 / D 1

p

1 2 t

u2

e  2t du;

1; x1  x10 , 0; x1 < x10 ,

is the Heaviside function. The transition density is obtained by differentiating the previous transition density: @2 Ft .x1 ; x2 j x10 ; x20 / @x1 @x2 Z x2 x20 0 @ @ 1 u2 x1 Hx 0 .x1 / p e  2t du D @x1 1 @x2 1 2 t  2 x2 x 0 1 2  1 2t x0 1 D ıx 0 .x1 / 0 p : e 1 x1 2 t

pt .x0 ; x/ D

Hence the heat kernel of A D 12 x12 @2x2 is given by 1  2t 1 K.x0 ; xI t/ D ıx 0 .x1 / ˝ 0 p e 1 x1 2 t



0 x2 x2 0 x 1

2 ;

for any initial point .x10 ; x20 / with x10 6D 0. If x10 D 0, the transition distribution is   FXjX0 .x1 ; x2 j 0; x20 / D P X1 .t/  x1 ; X2 .t/  x2 j X1 .0/ D 0; X2 .0/ D x20 D P .0  x1 ; x20  x2 / D H0 .x1 / Hx 0 .x2 /; 2

so by taking the derivative, we get pt .0; x/ D ı0 .x1 /ıx 0 .x2 /: 2

8.19 Grushin’s Operator

191

8.19 Grushin’s Operator The Grushin operator G D

1 2 .@ C x12 @2x2 / 2 x1

  is a sub-elliptic operator which is the generator for the Ito diffusion X1 .t/; X2 .t/ given by dX 1 .t/ D dW 1 .t/; dX 2 .t/ D X1 .t/ dW 2 .t/; with X1 .0/ D x10 , X2 .0/ D x20 and W1 .t/, W2 .t/ one-dimensional independent Brownian motions starting at zero. Solving yields X1 .t/ D x10 C W1 .t/; Z t X1 .s/ dW 2 .s/; X2 .t/ D x20 C 0

and the heat kernel of G is given by the following transition probability (see item 6 of Sect. 8.6): pt .x0 ; x/ D EŒı.x1  X1 .t//ı.x2  X2 .t//   Z 1 D E ı.x1  X1 .t// e i.x2 X2 .t // d 2 Z i h Rt 1 0 D e i.x2 x2 / E ı.x1  X1 .t//e i 0 X1 .s/dW 2 .s/ d: (8.19.76) 2 The following idea of computing comes from [31]. The expectation operator E is the Wiener integral over W1 .t/ and W2 .t/. Since X1 .t/ is independent of W2 .t/, we may integrate first with respect to W2 .t/ and get h Rt i R 1 2 t 2 EW2 e i 0 X1 .s/ dW 2 .s/ D e  2  0 X1 .s/ ds ; see (8.6.33). Substituting back in (8.19.76) yields 1 pt .x0 ; x/ D 2 D

1 2

Z Z

0

1

e i.x2 x2 / EŒı.x1  X1 .t//e  2  0

e i.x2 x2 / v.x10 ; x1 ; tI / d;

2

Rt

0

X12 .s/I ds

 d (8.19.77)

192

8 The Stochastic Analysis Method

where here E denotes the expectation with respect to W1 .t/ and h i R 1 2 t 2 v D v.x10 ; x1 ; tI / D E ı.x1  X1 .t//e  2  0 X1 .s/ ds i h R 1 2 t 0 2 D E ı.x1  .x10 C W1 .t///e  2  0 .x1 CW1 .s// ds : From the Feynman–Kac formula(see Theorem 8.23.1), with the quadratic potential V .x/ D 12 2 x 2 , the element v verifies the initial boundary problem @t v D

1 1 @ 2  2 x12 v; 2 x1 2

vjt D0 D ıx0 .x1 /: 1

From Theorem 3.16.1, we get vD p

s

1 2 t

t t 1 e 2t sinh.t / sinh.t/

h

x12 C.x10 /

2



cosh. t /2x1 x10

i

; t > 0:

Substituting back (8.19.77) yields 1 1 pt .x0 ; x/D p 2 2 t

Z s

1 t i.x2 x20 / 2t e sinh.t/

t sinh.t / Œ

  2 x12 C.x10 / cosh. t /2x1 x10 

d;

which, after the substitution  D t, becomes 1 pt .x0 ; x/ D .2 t/3=2

Z r

 1 e t f .x0 ;x;t I / ; sinh 

with f .x0 ; x; tI / D i.x2  x20 / 

 i  h 2 2 x1 C .x10 / coth   2x1 x10 sech : 2

8.20 Squared Bessel Process The operator 1 2 x > 0; x@ C @x ; 2 x is the generator of the following squared Bessel process: dX t D dt C

p Xt =2 dW t ;

t  0;

(8.20.78)

8.22 Limitations of the Method

193

with  > 1=8. The heat kernel of (8.20.78) with zero boundaries at x D 0; 1 is given by the transition density of the aforementioned process in terms of the modified Bessel function(see [29]): K.x0 ; xI t/ D

p e .xCx0 /=.t =4/  x 2 1=2 I4 1 .8 xx0 =t/; t=4 x0

x; x0 > 0; t > 0: (8.20.79)

8.21 CIR Processes The operator 1 2 x@ C .0  1 x/@x ; 2 x

x > 0;

(8.21.80)

is the generator of the CIR process6 dX t D .0  1 x/dt C

p Xt =2 dW t ;

t > 0;

with 0 ; 1 > 0, constants. These types of processes have been used to model stochastic interest rates in finance: see [37]. The heat kernel of the operator (8.21.80), given as the transition density of the CIR process, is given for instance in [29]: !2 0  12 xe 1 t Cx0 41 e 1 t x 1 t 4

K.x0 ; xI t/ D t e e 1 e 1 t 1 e 1  1 x0  8 p  1 I4 0 1 t xx0 e 1 t ; e 1 1 x0 ; x > 0, t > 0.

8.22 Limitations of the Method The stochastic method presented in this chapter works for a large class of operators. However, it does not apply to all operators. There are several limitations of the method that will be discussed in the following. These limitations can be overcome by using some of the methods presented in the other chapters.

6

CIR stands for Cox, Ingersoll and Ross.

194

8 The Stochastic Analysis Method

1. Explicit solution but the transition density is hard to get. Consider the case of the one-dimensional linear noise with drift given by dX t D r dt C ˛Xt dW t ;

r; ˛ 2 R:

(8.22.81)

Given x0 2 R, Theorem 8.2.2 ensures the existence and uniqueness of a process Xt satisfying (8.22.81) with the initial condition X0 D x0 . According to Theorem 8.3.1, the generator associated with the process Xt is given by AD

@ 1 2 2 @2 Cr : ˛ x 2 2 @x @x

(8.22.82)

Without loss of generality, we may assume ˛ D 1. Equation (8.22.81) becomes dX t  Xt dW t D r dt: 1

Multiplying by the integrating factor t D e Wt C 2 t yields t dX t  t Xt dW t D rt dt:

(8.22.83)

We shall show that the left side is exact. Using the product rule and Ito’s formula yields dt D t dW t C t dt; dt dX t D t Xt dt; since dt2 D dt dW t D dW t dt D 0 and dW t dW t D dt. Then d.t Xt / D dt Xt C t dX t C dt dX t D t Xt dW t C t Xt dt C t dXt  t Xt dt D t dX t  t Xt dW t ; which is the left side of (8.22.83). Integrating in d.t Xt / D rt dt yields Z t Xt D 0 X0 C r

t

s ds ” Z t 1 Xt D 1 x C r s ds ” 0 t t 0 Z t 1 1 1 Xt D e Wt  2 t x0 C re Wt  2 t e Ws C 2 s ds; 0

0

(8.22.84)

8.23 Operators with Potential and the Feynman–Kac Formula

195

where 0 D 1 and X0 D x0 . If r D 0, then Xt has a lognormal distribution and this case was treated in Sect. 8.10. However, in the case r 6D 0, the transition density of the process (8.22.84) is not easy to obtain. 2. The stochastic differential equation has a unique solution, but it cannot be solved explicitly. Consider the following equation: dX t D dt C .cos Xt / dW t ; X0 D x0 : It is obvious that the existence and uniqueness condition of Theorem 8.2.2 applies and the equation has a unique solution Xt . However, a closed-form solution is hard to obtain. Hence the stochastic method is not suitable for finding the heat kernel of the operator AD

d 1 d2 .cos x/2 2 C : 2 dx dx

3. The stochastic differential equation might have multiple solutions. This is the case of equation 1=3

dX t D 3Xt

2=3

dt C 3Xt

dW t ;

(8.22.85)

X0 D 0; considered in Sect. 8.2. A direct application of the algorithm in Sect. 8.5 will not produce the heat kernel for AD

9 4=3 d 2 d x C 3x : 2 2 dx dx

The aforementioned operator is singular at x D 0 and each solution of (8.22.85) has a contribution to the kernel that starts at the origin.

8.23 Operators with Potential and the Feynman–Kac Formula In the previous sections we computed the heat kernel for a number of operators of the form X 1X @2 @ AD . T /ij C bk (8.23.86) 2 @xi @xj @xk i;j

k

196

8 The Stochastic Analysis Method

associated with the n-dimensional Ito diffusion process dX t D b.Xt / dt C .Xt / dW.t/;

(8.23.87)

which is supposed to satisfy the existence and uniqueness conditions of Theorem 8.2.2. In this section we deal with an extension of Kolmogorov’s backward equation (see Theorem 8.4.1), for operators of the type A  V .x/, where V .x/ is a potential function. The first formula of this type deals with the n-dimensional Laplacian; it was found by Kac [75] in 1949. The result states that the equation @v 1 D v  V .x/v; @t 2

t > 0;

v.0; x/ D f .x/; has the solution given by h i Rt v.t; x/ D E x f .Xt / e  0 V .Xs / ds ;

(8.23.88)

where E x is the conditional expectation given that the process Xt is a Brownian motion starting at x; i.e., Xt D Wt C x with W0 D 0. The extension of the aforementioned result to any operator of the type (8.23.86) is given below. For a proof the reader can consult the references [83, 95]. Theorem 8.23.1 (Feynman–Kac formula). Let Xt be the continuous solution of the stochastic differential equation (8.23.87) and let A be its generator. Let f 2 C02 .Rn / and V .x/ be lower-bounded continuous functions on Rn . Then the function v.t; x/ D E x Œf .Xt / e 

Rt

0

V .Xs / ds

;

t  0; x 2 Rn ;

(8.23.89)

satisfies the Cauchy problem @v D Av  V v; @t

t > 0;

v.0; x/ D f .x/: Here E x denotes the conditional expectation given that X0 D x. We note that if V D 0, then the Feynman–Kac formula becomes the Kolmogorov forward equation result; see Theorem 8.4.1. In spite of its simplicity, formula (8.23.89) cannot be worked out explicitly unless the potential V .x/ is of a very particular form. Its importance is more of a theoretical

8.23 Operators with Potential and the Feynman–Kac Formula

197

value, since it provides an integral representation of the heat equation even in cases that do not have closed-form solutions. For example, in the case of quartic potential, the solution of the equation 1 @v D @2x v  x 4 v; @t 2 v.0; x/ D f .x/ is v.t; x/ D E x Œf .Wt C x/ e 

Rt

4 0 .Ws Cx/

t > 0;

ds

;

t  0; x 2 R:

It is worth noting that the Feynman–Kac formula can also be expressed in terms of path integrals; see Chap. 7.

Part II

Heat Kernel on Nilpotent Lie Groups and Nilmanifolds

Chapter 9

Laplacians and Sub-Laplacians

9.1 Sub-Riemannian Structure and Heat Kernels We have seen in the previous chapters how an elliptic operator can be associated in a natural way with a geometric Riemannian structure. In a similar way sub-elliptic operators arise from similar structures, called sub-Riemannian structures, which will be discussed next. References for sub-Riemannian manifolds are [27] and [92].

9.1.1 Sub-Riemannian Structure Let M be a smooth manifold. We shall assume M without boundary throughout this chapter. A sub-bundle H of the tangent bundle T .M / is called nonholonomic if the vector fields of the sub-bundle H and a finite number of iterations of their Lie brackets span the whole tangent space at each point. The sub-bundle H is sometimes called the horizontal distribution. Definition 9.1.1. A connected manifold M is called a sub-Riemannian manifold if it has a nonholonomic sub-bundle H with a positive definite, nondegenerate metric on H. The aforementioned nonholonomic property of the sub-bundle H is also called the bracket generating property or H¨ormander’s condition [65]. Sometimes it is also called Chow’s condition [35]. Next we provide the physical interpretation for the previous definition. The coordinates of a physical system can be represented as a point on a manifold, called the configuration space. The states of the system, which consist of pairs of coordinates and momenta, form the cotangent bundle of a manifold, called the phase space. The physical system is Riemannian if the states “can move in any direction”; i.e., there are curves in the configuration space starting at a given point pointing in all directions. These systems have the freedom to move in all directions. However, there are examples of mechanical systems with nonholonomic constraints which are not of the type just described. A rolling ball on a surface or a O. Calin et al., Heat Kernels for Elliptic and Sub-elliptic Operators, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-4995-1 9, c Springer Science+Business Media, LLC 2011 

201

202

9 Laplacians and Sub-Laplacians

penny rolling on a plane are two examples of these kind of systems. In this case the system is sub-Riemannian, which means that the structure moves only along horizontal directions, i.e., directions tangent to the sub-bundle H. Given an initial configuration and a final configuration, it is natural to ask whether it is always possible for a sub-Riemannian system to move from the initial position to the final position in the configuration space. Everyone has faced this problem while parking a car. Starting from a given position, the car has to move into a given final parking position. Since this motion cannot be done by only one smooth maneuver, the horizontal curve on the associated sub-Riemannian manifold will be just piecewise smooth. The following global connectedness result holds for subRiemannian manifolds: Theorem 9.1.2 (Chow’s theorem, [35]). Any two points of a sub-Riemannian manifold can be joined by a piecewise smooth horizontal curve, i.e., a curve tangent to the horizontal distribution H. Gromov showed that in some particular cases the “piecewise” attribute can be omitted, but in general it cannot. There are examples of physical systems which have states that cannot be connected by a horizontal curve. This is the case with thermodynamical systems where the horizontal curves correspond to adiabatic processes, i.e., processes along which there is no heat exchange. J. Charney was the first to note that there are always states which cannot be joined by an adiabatic curve, i.e., there are states which cannot be connected by a horizontal curve; see, for instance, [27], Chap. 3. We shall list below several basic definitions related to a sub-Riemannian structure. Definition 9.1.3. 1. The dimension dim T .M /=H is called the codimension of the sub-Riemannian structure H. 2. Let .M; T .M // be the space of all smooth vector fields on M; and set  0 .M; H/ D .M; H/, the space of smooth vector fields on M taking values in H. For any k 2 N, let  k .M; H/ D  0 .M; H/ C Œ 0 .M; H/;  k1 .M; H/: For any point x 2 M , we denote by x` .H/ the values of the tangent vectors in  ` .M; H/ at the point x 2 M: If for some k > 0, xk .H/ = Tx .M / but xk1 .H/ 6D Tx .M /, then we say the sub-Riemannian structure is of step k C 1 at the point x. The sub-Riemannian structure is step k on the domain U if it is step k at each point x 2 U: 3. If there is an ascending sequence of sub-bundles fH` g of T .M / H D H0  H1  H1      Hk D T .M /

9.1 Sub-Riemannian Structure and Heat Kernels

203

such that  ` .M; H/ D .M; H` /, then the sub-Riemannian structure is called regular. We have denoted by .M; H` / the space of smooth vector fields taking values in H` . Note that in this case all sub-bundles H` are nonholonomic. 4. A sub-Riemannian structure H is called minimal if there is no sub-bundle in H which satisfies the bracket generating property. Two additional conditions on the nonholonomic sub-bundle H shall be required throughout the book. (Sub-1) The horizontal sub-bundle H is trivial. This means that we can consider globally defined, nowhere-vanishing smooth vector fields fXi g, i D 1; : : : ; dim H, satisfying H¨ormander’s condition, which guarantees the hypoellipticity of the following second-order operator: dim XH

Xi  Xi :

i D1

The second assumption is (Sub-2) There is a volume form dVM with respect to which the vector fields fXi g are all skew-symmetric. If the aforementioned conditions hold, we say that the sub-Riemannian structure is “trivializable,” or is of the strong sense. It is worth noting that even if the typical examples of sub-Riemannian manifolds are contact manifolds, they are not always trivializable structures. Furthermore, the associated sub-elliptic operator sub D 

dim XH

Xi 2

i D1

is called the sub-Laplacian. Our next main purpose is to construct heat kernels for sub-Laplacians on sub-Riemannian manifolds. A special role will be played by the Grushin-type operators that will be treated in Sect. 10.3. Most of the operators in this chapter are sub-Laplacian operators on nilpotent Lie groups and their quotient spaces. It is a rather convenient assumption to consider the metric on the sub-bundle H in such a way that the vector fields fXi g are orthonormal.

9.1.2 Heat Kernel of the Sub-Laplacian and Laplacian Under the aforementioned conditions (Sub-1) and (Sub-2), the sub-Laplacian sub is symmetric and positive with respect to the volume form dM V , whose existence is assumed by the latter condition. If the metric is extended to the entire tangent

204

9 Laplacians and Sub-Laplacians

space, then the Laplacian is symmetric and positive with respect to the Riemannian volume form. The volume element that we assumed to exist for the sub-Riemannian manifold will coincide in this case with the Riemannian volume form, up to a multiplicative constant, as we shall see in the following examples. Moreover, we have Theorem 9.1.4 ([105]). If a sub-Riemannian metric on a noncompact manifold can be extended to a complete Riemannian metric, then the sub-Laplacian is essentially self-adjoint on the space of smooth functions with compact support. In all the cases treated in the sequel, we encounter left invariant operators on nilpotent Lie groups, for which the volume form dV M postulated by (Sub-2) is a Haar measure, and the conditions (Sub-1) and (Sub-2) are satisfied with respect to this measure. In this case, a left invariant sub-Riemannian metric defined on a left invariant nonholonomic sub-bundle can be extended to a left invariant complete metric on the entire space. This is the reason why in the following cases we shall not distinguish between the sub-Laplacian sub , respectively the Laplacian , both defined on C01 .M /, and their unique self-adjoint realizations on the space L2 .M; dM V /. Invoking the spectral decomposition theorem Z sub D

0

1

dE ;

Q g/ is the with fE g the spectral measure, we know that the heat kernel K.tI g; kernel distribution of the bounded operator e t sub D

Z 0

1

e t dE W L2 .M / ! L2 .M /;

t > 0:

This kernel is smooth because 1. sub is hypoelliptic 2. For any integer k, the operator subk ı e t sub D e t sub ı subk is defined  1  T on L2 .M / and e t sub maps continuously L2 .M / to domain of subk D kD1

C 1 .M / 3. e t sub can be extended to the entire space of distributions D0 .M / on M from L2 .M /. We have e t sub D e t =2sub ı e t =2sub as a map from D0 .M / to C 1 .M / Of course, the previous arguments are also valid for the case of Laplacians and the heat kernel for Laplacians on C 1 .RC  M  M /: We shall next review the purpose of our future exposition. (1) We are interested in the spectral decomposition of the sub-Laplacian and Laplacian in a multiplication operator form. This means finding a measure space .X; dm/, a (positive) function ' on X, and a unitary transformation

9.1 Sub-Riemannian Structure and Heat Kernels

205



U W L2 .M / ! L2 .X; dm/ such that sub D U1 ı M' ı U; where M' W L2 .X; dm/ 3 f 7! 'f 2 L2 .X; dm/ is the multiplication operator by the function '. (2) We shall provide an explicit construction for the heat kernel K.tI x; y/ of the sub-Laplacian sub (and also Laplacian). This means finding a solution K.tI x; y/ for   @ sub C K.tI x; y/ D 0; @t

Z lim

t !0 M

K.tI x; y/f .y/dM V .y/ D f .x/:

In the case of nilpotent Lie groups, using the (left) invariance of the heat kernel K.tI g  x; g  y/ D K.tI x; y/;

8g; x; y 2 M;

it follows that the heat kernel K.tI x; y/ is of the form K.tI x; y/ D kt .y 1  x/; with kt .x/ a smooth function on RC  M: (3) We shall find explicit expressions for the Green functions and fundamental solutions of the operators sub and . (4) We aim to obtain explicit expressions for the heat kernel of the Grushin-type operators defined in Sect. 10.3. There are many papers in the literature treating the hypoelliptic operators on nilpotent Lie groups (for instance, [15, 100] and papers cited therein). However, our concern here is to construct the heat kernel in terms of special functions. We shall use Hermite functions and hyperbolic functions for the two-step cases, while for the three-step cases the computation might be achieved by using elliptic functions and other special functions; see [21,22,58]. However, until now there are no explicit formulas for Laplacians and sub-Laplacians on nilpotent Lie groups with step greater than three (including the Engel group). Therefore, we are limited in our endeavor of finding- closed form formulas only for the two-step nilpotent Lie groups. Remark 9.1.5. If we have a spectral decomposition of  (or sub ) in the multiplication operator form, then the heat kernel is expressed as e t  D U ı e t M' ı U1 : This is similar to the case of Euclidean spaces where the operator U is the Fourier transform. So (1) leads to (2), but not the other way around.

206

9 Laplacians and Sub-Laplacians

Integrating the heat kernels, we obtain Green functions. If the heat kernel is K.tI x; y/ 2 C 1 .RC  M  M /; then the Green function G.x; y/ can be expressed as Z M

Z G.x; y/f .y/dM V .y/ D

0

1

Z G

K.tI x; y/f .y/dM V .y/ ^ dt:

Even if we have an explicit expression for the heat kernel, this does not provide the spectral decomposition of the (sub-)Laplacian in a multiplication operator form directly. For this purpose we are still required to find a measure space .X; dm/, a unitary transformation between L2 .G/ and L2 .X; dm/, and a positive function on X, with respect to which the Laplacian (sub-Laplacian) can be expressed as a multiplication operator.

9.1.3 The Volume Form In the definition of the sub-Riemannian manifold we have assumed that there exists a volume form dM V .x/ such that each vector field Xk is skew-symmetric with respect to the inner product induced by this volume form: Z

Z M

Xk .f /.x/g.x/dM V .x/ D 

M

f .x/Xk .g/.x/dM V .x/;

f or g 2 C01 .M /:

For regular sub-Riemannian manifolds, we can construct a volume form in an intrinsic way (see [92] for Popp’s measure) based on the inner product defined on the nonholonomic sub-bundle, even if the vector fields taking values in the nonholonomic sub-bundle are not necessarily skew-symmetric. For the sake of simplicity, in this section we shall explain the construction of the volume form just for the case of two-step regular sub-Riemannian manifolds. Let X 2 Hx and Y 2 Hx be tangent vectors which are extended to the vector fields XQ and YQ on M taking values in H. Then the value of the bracket ŒXQ ; YQ  2 Tx .M / (mod Hx ) does not depend on the extensions XQ and YQ . Thus we have a well-defined bundle map  W H ˝ H ! T .M /=H: The map  is surjective by the bracket generating assumption .M; H/ C Œ.M; H/; .M; H/ D .T .M //: Now we can introduce a metric on T .M /=H such that it is isometric with the orthogonal complement of the kernel of the map . Note that H ˝ H has a tensorial

9.2 Laplacian and Sub-Laplacian on Nilpotent Lie Groups

207

product metric from H. Then we have an exact sequence of vector bundles with metrics except total space T .M /: 0 ! H ! T .M / ! T .M /=H ! 0:

(9.1.1)

Since the isomorphism dim ^H

H ˝

dim M^ dim H

 T .M /=H

dim ^M

Š

T  .M /

does not depend on the choice of the splitting of the previous exact sequence, with dim VM  the help of the metrics on H and T .M /=H, we can trivialize T .M / by choosing dual orthonormal bases of these two bundles at each point. This provides a volume form on M: Remark 9.1.6. In the following examples the vector fields trivializing a nonholonomic sub-bundle are skew-symmetric with respect to the volume form constructed by this way (up to a multiplicative constant).

9.2 Laplacian and Sub-Laplacian on Nilpotent Lie Groups In this section we shall explain the basic properties of nilpotent Lie groups and subRiemannian structures on such Lie groups. We shall also deal with several examples of nilpotent Lie groups and sub-Laplacians on them for which we shall construct heat kernels in later sections.

9.2.1 Nilpotent Lie Groups Let G be a nilpotent Lie group with the Lie algebra g. This means that if we put g1 D Œg; g, g2 D Œg; g1 , : : :, gk D Œg; gk1 , : : :, then there exists N 2 N such that gN D f0g. For an elementary treatment of nilpotent Lie groups, we refer the reader to [110]. We begin by reminding the reader of one of the most fundamental results. Theorem 9.2.1. Let G be a connected and simply connected nilpotent Lie group. Then the exponential map exp W g ! G is a diffeomorphism. Because of this fact, we can work on the linear space g using the Campbell– Hausdorff formula exp X  exp Y D exp.X C Y C 1=2ŒX; Y  C 1=12ŒŒX; Y ; Y  X  C : : :/I X; Y 2 g;

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9 Laplacians and Sub-Laplacians

which says that we can introduce the group multiplication in the Lie algebra g  through the identification exp W g ! G. The group law is defined on g as in the following: If exp X  exp Y D exp Z; then we define X  Y D Z; with Z given by Z D X C Y C 1=2ŒX; Y  C 1=12ŒŒX; Y ; Y  X  C : : : (finite sum): Q : : : ; / 2 G Š g if we consider them We shall use the notations g; h; : : : (or g; Q h; as elements of the Lie group g, and X; Y; : : : 2 g if we consider them as elements of the Lie algebra g. This way we eliminate the confusion of whether we are dealing with g as a group or as a Lie algebra. e the left invariant vector field on the Lie group G Let X 2 g, and denote by X defined by e .f /.g/ D df .g  exp tX / X ; dt jt D0 e , we have the following where f 2 C 1 .g/. By the correspondence g 3 X ! X trivialization of the tangent bundle T .G/ of the group G: G  g ! T .G/; e g 2 Tg .G/: .g; X / 7! X We also have a trivialization of the cotangent bundle T  .G/: G  g ! T  .G/; .g; / 7! e  g 2 Tg .G/; where the left invariant one-form e  for  2 g is defined by the formula e e g / D .X /; X 2 g:  g .X g Let fXi gdim i D1 be a basis of the Lie algebra g. We consider the coordinates on the Lie group G D g introduced by the correspondence

.x1 ; : : : ; xdim g / $ g D

dim Xg

xi Xi :

i D1

Then we can take the differential form dg D dx1 ^    ^ dxdim g as a Haar measure.

9.2 Laplacian and Sub-Laplacian on Nilpotent Lie Groups

209

e as a Proposition 9.2.2. Let X 2 g. If we consider the left invariant vector field X first-order differential operator on the group G, e W C 1 .G/ ! C 1 .G/; f 7! X e .f /; X then it is skew-symmetric with respect to the inner product defined by a Haar measure. Proof. Let '; Z

2 C01 .G/. We have

e .'/.g/  X G

  d' g  exp.tX /  .g/dg .g/dg D lim t !0 G dt jt D0   Z d g  exp.tX / D lim '.g/  dg t !0 G dt jt D0 Z e . /.g/dg; '.g/  X D Z

G

since the Haar measure is left and right invariant in the case of the nilpotent Lie groups.  Let 1 W g ! g=Œg; g be the projection map and let fXi g`iD1 be linearly independent elements such that f1 .Xi /g`iD1 spans the quotient space g=Œg; g. Then the subspace fXQ i g`iD1 defines a regular sub-Riemannian structure on the group G. If the group is a k-step nilpotent Lie group, then this structure is minimal, nonholonomic, and of step k.

9.2.2 The Heisenberg Group Let H3 be the three-dimensional Heisenberg group identified by R3 . The product law is given by the formula   R3  R3 3 .x; y; z/; .x; Q y; Q zQ/ 7! .x; y; z/  .x; Q y; Q zQ/   D x C x; Q y C y; Q z C zQ C 1=2.x yQ  yx/ Q 2 R3 : Then its Lie algebra h3 is identified by itself; i.e., the exponential map is the identity. H3 can also be realized as a real 3  3 matrix space 80 9 1 < 1 x z = H3 Š @0 1 y A I x; y; z 2 R : : ; 0 0 1 The identification is given by the map 0

1 1 x z C xy 2 H3 3 .x; y; z/ 7! @0 1 y A: 0 0 1

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9 Laplacians and Sub-Laplacians

Remark 9.2.3. There are also other ways to realize the group H3 as a matrix space. For example, if we let 0

0 B0 X DB @0 0

1 0 0 0

0 1 0 0 0 B C 0 0 0C ; Y DB @0 0 1A 0 0 0

0 0 0 0

0 1 0 0 0 B C 0 0 1C and Z D B @0 0 0A 0 0 0

1 0 0 0

0 0 0 0

1 2 0C C; 0A 0

we have ŒX; Y  D Z, and the space of matrices 80 0 ˆ ˆ > = 0 y C C I x; y; z 2 R > 0 x A > ; 0 0

is isomorphic to h3 . So the three-dimensional Heisenberg group H3 is identified by a subgroup in 4  4 matrix space as 80 1 ˆ ˆ > = 0 y C C I x; y; z 2 R : > 1 x A > ; 0 1

The exponential map is given by 0

0 B0 exp W h3 3 B @0 0

x 0 0 0

0 1 0 y z B C 0 0 y C 7! Id C B @0 0 x A 0 0 0

x 0 0 0

1 0 1 y z B C 0 y C B0 D 0 x A @0 0 0 0

x 1 0 0

1 y z 0 y C C 2 H3 : 1 x A 0 1

In the following we shall consider the three-dimensional Heisenberg group as R3 . Let X D .1; 0; 0/, Y D .0; 1; 0/ and Z D .0; 0; 1/ be the basis of the Lie e, Y e and Z e are given by algebra h3 . The left invariant vector fields X eD @ y @; X @x 2 @z eD @ Cx @; Y @y 2 @z eD @: Z @z

9.2 Laplacian and Sub-Laplacian on Nilpotent Lie Groups

211

A sub-Riemannian structure H on H3 in the strong sense is defined by a sube Y eg. We consider a left invariant metric on H such that the bundle spanned by fX; e and Y e are orthonormal at each point. In this case the sub-Laplacian vector fields X sub is e2 C Y e2 /: sub D .X (9.2.2) Furthermore, if we consider the left invariant Riemannian metric on H3 such that e, Y e and Z e are orthonormal at each point, then the metric tensor is given by X 0

1 y  xy 4 2 C 2 1 C x4  x2 A ;  x2 1

2

1C y B xy4 @ 4 y 2

with the inverse matrix

0

1 B @ 0  y2

0 1 x 2

 y2 1C

1

C x A: 2 x 2 Cy 2 4

Hence the Laplacian H3 is 

H3

   @2 @2 @2 @2 x 2 C y 2 @2 D C 2 y Cx C 1C @x 2 @y @x@z @y@z 4 @z2 e2  Y e2  Z e 2 D sub  Z e2 : D X

9.2.3 Higher-Dimensional Heisenberg Groups The higher-dimensional Heisenberg group is given by H2nC1 D R2nC1 D R2n  R Š Rn  Rn  R together with the following group law: 

  < x; yQ >  < y; xQ >  ; .x; yI z/; .x; Q yI Q zQ/  7 ! x C x; Q y C yI Q z C zQ C 2

for any .x; yI z/; .x; Q yI Q zQ/ 2 Rn  Rn  R. In the aforementioned formula, < ;  > denotes the standard inner product on Rn . H2nC1 is realized in the matrix space

H2nC1

80 ˆ 1 x1 x2 ˆ ˆ ˆB0 1 0 ˆ ˆ B ˆ ˆB ˆ    xn z > > C > >    0 y1 C > > C > >    0 y2 C = C n :: C I x; y 2 R ; z 2 R >  0 : C > C > > C > > > 1 yn A > > ;  0 1

212

9 Laplacians and Sub-Laplacians

by the map 0

H2nC1

1 x1 x2 B0 1 0 B B B0 0 1 B n n D R  R  R 3 .x; yI z/ 7! B B0 0 0 B B :: @: 0  

1    xn z C <x;y> 2 C  0 y1 C C  0 y2 C C :: C: C  0 : C C A 1 yn  0 1

The Lie algebra of H2nC1 , which can also be regarded as R2nC1 , is denoted by h2nC1 . For 1  i  n, the vectors Xi D .0; : : : ; 0 ; 1; 0; 0; : : : ; 0; I 0/; „ ƒ‚ … i 1 zero Yi D .0; : : : ; 0; 0; : : : ; 0; 1; 0; : : : I 0/; ƒ‚ … „ nCi 1 zero Z D .0; : : : ; 0; 0; : : : ; 0I 1/ ƒ‚ … „ 2n zero form a basis in the Heisenberg algebra h2nC1 . The corresponding left invariant vector fields are expressed as ei D @  X @xi ei D @ C Y @yi ei D @ : Z @z

yi 2 xi 2

@ ; @z @ ; @z

(9.2.3) (9.2.4) (9.2.5)

Now we have a left invariant sub-Riemannian structure H spanned by the left ei I i D 1; : : : ; ng. We shall consider the left invariei ; Y invariant vector fields fX ei ; Y ei I i D 1; : : : ; ng are orthonormal at each point ant metric on H such that fX of H2nC1 . The sub-Laplacian in this case is given by sub D 

n X

ei 2 : ei 2 C Y X

(9.2.6)

i D1

We may also consider the left invariant Riemannian metric g on H2nC1 such that ei ; Y ei I i D 1; : : : ; ng and Z e are orthonormal. The the left invariant vectors fields fX metric tensor in this case is given by

9.2 Laplacian and Sub-Laplacian on Nilpotent Lie Groups

213

    1 0  @ @ g @x@ ; @y@ g @x@ ; @z g @x ; @x@ j i j i B  i     C C B @ @ @ @ @ @ C g gij D B ; ; ; g g B @yi @xj @yi @yj @yi @z C @      A @ @ @ @ ; @x@ ; @y@ ; @z g @z g @z g @z j

0 2 1 C y41 B :: B : B B : B : B : B B yn y1 B 4 B DB B  B B B  B B B  B B @  

j

 :: :



:



::





  

1C

2 yn 4







x y  i4 j

xi yj 4

 yi 2



yi yj 4

yj yi 4



y1 yn 4



 



 :: : 

 1C

yj xi 4



x12



xi yj 4



:: :



 :: :



xj xi 4

::

 

  x2i

 

4

1



xi xj 4

: x2

1 C 4n 

C C C C C C C C C C x1 C : 2 C C C  x2i C C C C C :: C : A 1 yi 2

Then the Laplacian H2nC1 is e2: H2nC1 D sub  Z In terms of coordinates .x1 ; : : : ; xn ; y1 ; : : : ; yn ; z/, the Laplacian and sub-Laplacian can be expressed by  X X  @2 X @2 @2 @2 C C y  x i i @xi 2 @yi 2 @xi @z @yi @z !  P 2 x i C yi 2 @2 ; (9.2.7)  4 @z2  X X  @2 X @2 @2 @2 D C C y  x i i @xi 2 @yi 2 @xi @z @yi @z !  P 2 xi C yi 2 @2 : (9.2.8)  1C 4 @z2

sub D 

H2nC1

9.2.4 Quaternionic Heisenberg Group Let H be the field of quaternions over the real numbers with the basis f1; i; j; kg. Their products are given by the following usual relations:

214

9 Laplacians and Sub-Laplacians

1i D i1 D i; 1j D j1 D j; 1k D k1 D k; i2 D j2 D k2 D 1; ij D ji D k; jk D kj D i; ki D ik D j; which obviously satisfy the associativity but not the commutativity. The elements of H are denoted by h D x0 1 C x1 i C x2 j C x3 k, with real coefficients x0 ; x1 ; x2 ; x3 2 R. The “conjugate”of h is defined by h D x0 1  x1 i  x2 j  x3 k. Then we have hk D k h; 8h; k 2 H: The norm, or the absolute value, of the element h 2 H is denoted by jhj and is given by r p X xi 2 : jhj D hh D We shall now consider the antisymmetric bilinear form Im.h; k/ D

 1 hk  kh ; 2

which introduces a Lie bracket on the vector space H ˚ R3 by H  H ! R3 Q 7! Œh; h Q D Im.h; h/: Q .h; h/ Then the space ˚  qh7 D H ˚ R3 D x0 1 C x1 i C x2 j C x3 kI z1 ; z2 ; z3 D .x0 ; x1 ; x2 ; x3 I z1 ; z2 ; z3 / j xi ; zi 2 R



becomes a two-step nilpotent Lie algebra with respect to the aforementioned bracket. We shall introduce a group multiplication on qh7 by   H ˚ R3  H ˚ R3 3 .h; z/; .k; zQ/ 7! .h; z/  .k; zQ/;   .h; z/  .k; zQ/ D h C k ˚ .z C zQ C Im.h; k/=2/ : We shall denote by qH7 the Lie group with this multiplication law. Let X0 D .1; 0; 0; 0I 0; 0; 0/ D 1 ˚ 0 2 H ˚ R3 ; X1 D .0; 1; 0; 0I 0; 0; 0/ D i ˚ 0 2 H ˚ R3 ; X2 D .0; 0; 1; 0I 0; 0; 0/ D j ˚ 0 2 H ˚ R3 ; X3 D .0; 0; 0; 1I 0; 0; 0/ D k ˚ 0 2 H ˚ R3

9.2 Laplacian and Sub-Laplacian on Nilpotent Lie Groups

215

be elements in qh7 which, together with their bracket generate the entire space qh7 . The expressions of the corresponding left invariant vector fields are given by 3

X @ e0 D @ C 1 X xi ; @x0 2 @zi

(9.2.9)

i D1

e 1 D @ C x0 @ C x3 @  X @x1 2 @z1 2 @z2 @ @ x x 3 0 @ e2 D X  C C @x2 2 @z1 2 @z2 e 3 D @ C x2 @  x1 @ C X @x3 2 @z1 2 @z2

x2 2 x1 2 x0 2

@ ; @z3 @ ; @z3 @ : @z3

(9.2.10) (9.2.11) (9.2.12)

We shall introduce on qH7 a sub-Riemannian structure H in the strong sense, i.e., a sub-bundle spanned by the left invariant vector fields fXQ i g3iD0 . The sub-Laplacian in this case is sub D 

3 X

ei 2 X

i D0

 3  2 X @2 @2 @2 @2 @ D C x  x C x  x i 0 3 2 @x0 @zi @x1 @z1 @x1 @z2 @x1 @z3 @xi2 i D0 @2 @2 @2 @2 @2  x0  x1  x2 C x1 @x2 @z1 @x2 @z2 @x2 @z3 @x3 @z1 @x3 @z2 ! 3 3 2 2 X X @ 1 @  x0  xi2 : (9.2.13) 2 @x3 @z3 4 @z i i D1 i D0 C x3

When considering a Riemannian metric on the entire space such that the vector ei g and fZ ej g are orthonormal at each point, this metric can be regarded as fields fX an extension of the metric defined on the sub-Riemannian structure. Consequently, the Laplacian in this case is given by  D sub 

@2 @2 @2  2  2: 2 @z1 @z2 @z3

Higher-dimensional quaternionic Heisenberg algebras qh4nC3 Š Hn ˚ R3 are defined by   .h1 ; : : : ; hn I z/; .hQ 1 ; : : : ; hQ n I zQ/

! n X 1 Im.hi ; hQ i / ; 7 ! h1 C hQ 1 ; : : : ; hn C hQ n I z C zQ C 2 i D1

216

9 Laplacians and Sub-Laplacians

  for all .h1 ; : : : ; hn I z/; .hQ 1 ; : : : ; hQ n I zQ/ 2 Hn ˚ R3 , and with notation Im hi ; hQ i D  hei hN i /. The Lie group associated with the Lie algebra qh4nC3 will be denoted by qH4nC3 , and it will be called the .4n C 3/-dimensional quaternionic Heisenberg group. 1 .h e 2 i hi

9.2.5 Heisenberg-Type Lie Algebra Let n and z be two real vector spaces endowed with the inner products < ;  >n and < ;  >z , respectively. We shall assume that there is a bilinear form J W z  n ! n such that (HT-1) W (HT-2) W

J.Z; J.Z; X // D  < Z; Z >z X; < J.Z; X /; Y >n D  < X; J.Z; Y / >n ;

(9.2.14) 8X 2 n; Z 2 z: (9.2.15)

The bilinear form J is sometimes called Clifford multiplication. Now we will define the skew-symmetric bilinear map B W n  n ! z given by B.X; Y / W z ! R;

B.X; Y /.Z/ D< J.Z; X /; Y >n :

Proposition 9.2.4. If Z1 and Z2 2 z are orthogonal, then J.Z1 ; J.Z2 ; Y // C J.Z2 ; J.Z1 ; X // D 0: Proof. Using the bilinearity of the map J; we have J.Z1 C Z2 ; J.Z1 C Z2 ; X // D J.Z1 ; J.Z1 ; X //CJ.Z1 ; J.Z2 ; X //CJ.Z2 ; J.Z1 ; X //CJ.Z2 ; J.Z2 ; X // D  < Z1 ; Z1 >z X CJ.Z1 ; J.Z2 ; X //CJ.Z2 ; J.Z1 ; X //  < Z2 ; Z2 >z X D  < Z1 C Z2 ; Z1 C Z2 ; >z X D  < Z1 ; Z1 >z X  < Z2 ; Z2 >z X; and hence J.Z1 ; J.Z2 ; X // C J.Z2 ; J.Z1 ; X // D 0:  Consider the inner product < ;  >z on z induced naturally by the inner product on z. We have the following result.

9.2 Laplacian and Sub-Laplacian on Nilpotent Lie Groups

217

Proposition 9.2.5. Let ' 2 z and assume that for any fixed X 6D 0, < B.X; Y /; ' >z D 0;

8 Y 2 n:

Then ' = 0. Proof. Let fZk g be an orthonormal basis on z and denote by fZk  g the dual basis on z . Then X < B.X; Y /; ' >z D < J.Zk ; X /; Y >n ak D 0; P where we expressed ' D ak Zk  . If we take Y D J.Zj ; X /, then by property (9.2.15) and Proposition 9.2.4, we have < B.X; Y /; ' >z D D

X X

< J.Zk ; X /; Y >n ak < J.Zk ; X /; J.Zj ; X / >n  < Zj ; Zj >z aj

k6Dj

D

X

< J.Zj ; X /; J.Zk ; X / >n  < Zj ; Zj >z aj D 0:

k6Dj

This implies that aj D 0: Hence ' D 0.



Corollary 9.2.6. For any fixed nonzero X 2 n, the mapping n ! z ; Y 7! B.X; Y / is surjective. We now define a two-step nilpotent Lie algebra structure on V D n ˚ z (or on n ˚ z by identifying z Š z) in such a way that for X; Y 2 n; ŒX; Y  D B.X; Y /; and all other brackets are zero. By Proposition 9.2.5, we have Proposition 9.2.7.

ŒV; V  D z D center:

This induces a two-step nilpotent Lie group structure on V given by V  V ! V;     1 .x; w/; .x; Q w/ Q 7! .x; w/  .x; Q w/ Q D x C x; Q w C wQ C B.X; Y / : 2

218

9 Laplacians and Sub-Laplacians

Remark 9.2.8. 1. Let C `Q .z/ denote the Clifford algebra with respect to the positive bilinear form Q.z/ D< z; z >z . This means that if T .z/ is the tensor algebra 3

T .z/ D R ˚ z ˚ z ˝ z˚ ˝ z ˚    ; and IQ is a two-sided ideal in T .z/ generated by the elements of the form z ˝ z C Q.z/; then C `Q .z/ is the quotient algebra T .z/=IQ : Conditions (9.2.14) and (9.2.15) required on the bilinear map J say that the vector space n has a Clifford module structure of the Clifford algebra C `Q .z/. 2. The Heisenberg-type Lie algebra was first defined in [77], where the Lie algebra g is two-step and is equipped with an inner product < ;  > such that for any Z 2 Œg; g z D center, the map J.Z/ W z? ! z? defined by (Ka-1): < J.Z/.X /; Y >D< Z; ŒX; Y  >D< ad.X / .Z/; Y >

(9.2.16)

satisfies (Ka-2): J.Z/2 D  < Z; Z > Id:

(9.2.17)

We have constructed a Heisenberg-type Lie algebra from the two vector spaces n and z satisfying conditions (HT-1) and (HT-2). Conversely, let’s assume that g satisfies conditions (Ka-1) and (Ka-2). Set n D z? D Œg; g? , and define the bilinear map J W z  z? ! z? by J.Z; X / D J.Z/.X /. Then it can easily be seen that z, z? and the bilinear map J satisfy conditions (HT-1) and (HT-2). 3. Heisenberg algebras are particular types of Heisenberg-type Lie algebras. Also, quaternionic Heisenberg algebras are Heisenberg-type Lie algebras. The latter case can be shown as in the following. Let H0 D fh 2 HI z D zg and consider the map H0  H ! H .z; h/ 7! z  h: Properties (9.2.14) and (9.2.15) are satisfied. The Heisenberg-type Lie algebra defined by this bilinear map is the quaternionic Heisenberg Lie algebra. Higherdimensional quaternionic Heisenberg Lie algebras can be constructed in a similar way. 4. All the Heisenberg-type Lie algebras are realized in terms of the Clifford module following periodicity of Clifford algebras and modules [5]. We consider next a sub-Riemanian structure on the Heisenberg-type Lie group V D n ˚ z .

9.2 Laplacian and Sub-Laplacian on Nilpotent Lie Groups

219

d For that purpose we choose arbitrary orthonormal bases fXi gN i D1 and fZk gkD1  of n and z , respectively, where N D dim n and d D dim z. These bases induce coordinates on the Lie group V D n ˚ z by

RN Cd 3 .x1 ; : : : ; xN I z1 ; : : : ; zd / $

X

xi Xi C

X

zj Zj 2 n ˚ z :

Then under the identification T .V / Š V  V; we consider a sub-Riemannian struce i gN given by ture H spanned by the left invariant vector fields fX i D1 H D V  n  V  V Š T .V /: Considering the constants of structure given by ŒXi ; Xj  D B.Xi ; Xj / D the left invariant vector fields are

P

Cikj Zk ,

X @ k ei D @  1 Ci;j xj ; X @xi 2 @zk jk

and hence the sub-Laplacian sub is given by sub D 

X

ei 2: X

Remark 9.2.9. According to a theorem in [55], we can choose an orthonormal basis in the Heisenberg-type Lie algebra V D n ˚ z with respect to which the structure constants are one of f1; 0; 1g. In this case, the expressions for the Laplacian and sub-Laplacian become very simple.

9.2.6 Free Two-Step Nilpotent Lie Algebra 1

N 2 N.N 1/ , 1  i < Let fXi gN i D1 be a basis of R . Consider also a basis fZi j g of R j  N . Define the Lie bracket relations by

ŒXi ; Xj  D ŒXj ; Xi  D

Zij ; for 1  i < j  N; 0; otherwise.

Then we can introduce a Lie group structure on RN.N C1/=2 D RN ˚ RN C2 by       N R ˚ RN C2  RN ˚ RN C2 3 x ˚ z; xQ ˚ zQ 0 1 X     1 7! x C xQ ˚ @z C zQ C xi xQ j  xj xQ i Zij A : 2 1i <j N

This group is called a free two-step nilpotent Lie group.

220

9 Laplacians and Sub-Laplacians

Let g be a two-step nilpotent Lie algebra. Consider the bases fXi gN i D1 and d fZj gj D1 for the complement of the derived algebra Œg; g and the derived algebra Œg; g, respectively. Let the structure constants fCi kj g be given by ŒXi ; Xj  D

X

Ci kj Zk :

k

Now let fgN CC 2 .CN2 D N.N  1/=2/ be the free two-step nilpotent Lie algebra N with the basis fXi ; Zi j g1i <j N such that ŒXi ; Xj  D Zi j : We define a Lie algebra homomorphism  by



X

 W fgN CC 2 ! g; N  X X X xi Xi C xi Xi C zi j Zi j D zi j Ci kj Zk :

Then we have the Lie group homomorphism between corresponding simply connected nilpotent Lie groups. In this sense, any two-step nilpotent Lie group (Lie algebra) is covered by a free two-step nilpotent Lie group (Lie algebra).

9.2.7 The Engel Group As we explained in Sect. 9.1.2, no one has been successful yet in expressing the heat kernel of a sub-Laplacian (or Laplacian) on nilpotent Lie groups of step greater than or equal to 3 in an explicit (integral) form. The explicit formula is missing even in the minimal dimensional case (dimension 4) nilpotent Lie group of step 3, called the Engel group. This group appears in various contexts and we shall also introduce it here, although we shall not give its heat kernel (for the Laplacian or sub-Laplacian) in any form. The crucial reason for which an explicit form of the heat kernel is missing is that we cannot solve the quartic oscillator in terms of any special functions, such as is done with the harmonic oscillator. Since the Engel group has codimension two and a three-step sub-Riemannian structure, we shall present here two sub-Laplacian operators and the Laplacian and note the difference from the two-step cases. Later, at the end of Sect. 10.3.3, we shall also express what the higher-step Grushin operators look like. Let e4 be a four-dimensional Lie algebra with the basis fX; Y; W; Zg with the following nonzero bracket relations: ŒX; Y  D W;

ŒX; W  D Z:

(9.2.18)

9.2 Laplacian and Sub-Laplacian on Nilpotent Lie Groups

221

Let E4 (identified with R4 ) be the Lie group whose Lie algebra is e4 . Then the group multiplication is given by the formula  1 .x; y; w; z/  .x; Q y; Q w; Q zQ/ D x C x; Q y C y; Q wCw Q C .x yQ  y x/ Q ; z C zQ 2  1 1 Q C .x  x/ Q .x yQ  y x/ Q : C .x wQ  wx/ 2 12 This group is realized in 4  4 real matrix space as 80 ˆ 1 ˆ ˆ z > > = C wC C I x; y; z; w 2 R : > yA > > ; 1

The identification is given by the map 0 1 B B0 .x; y; w; z/ 7! B @0 0

x 1 0 0

x2 2

x 1 0

2

z C xw C xy 2 3Š w C xy 2 y 1

1 C C C: A

If A 2 e4 is an element of Engel’s algebra, denote by AQ the left invariant vector field on the group E4 defined by   Q /.g/ D d f .g  exp tA/ A.f ; jt D0 dt

f 2 C 1 .E4 /:

Let H2 D spanfXQ ; YQ g. By the bracket relations (9.2.18), we obtain that H2 is a three-step, regular, minimal, codimension-2 sub-Riemannian structure on E4 in the Q YQ ; WQ g, we obtain a codimension-1, two-step strong sense. If we let H1 D spanfX; sub-Riemannian structure on E4 in the strong sense. The vector fields XQ , YQ , WQ , ZQ are expressed by @ @ @ y w ; XQ D @x @w @z @ @ YQ D Cx ; @y @w @ @ WQ D Cx ; @w @z @ ZQ D ; @z

222

9 Laplacians and Sub-Laplacians

while the sub-Laplacians associated with the sub-Riemannian structures H1 and H2 are given by .2/

sub D XQ 2  YQ 2 D

2 2 @2 @2 @2 2 2 @ 2 @   .x C y /  w C 2y @x 2 @y 2 @w2 @z2 @x@w

C 2w .1/

@2 @2 @2  2yw  2x ; @x@z @w@z @y@w

.2/

sub D sub  WQ 2 D

2 @2 @2 @2 @2 2 2 2 2 @   .x C y C 1/  .w C x / C 2y @x 2 @y 2 @w2 @z2 @x@w

C 2w

@2 @2 @2  2.yw C x/  2x : @x@z @w@z @y@w

If we fix a left invariant Riemannian metric on the Engel group E4 in such a way e; Y e; W e ; Zg e are orthonormal at each point, then the Laplacian  is given by that fX the formula 2 .1/ Q2 Q2  D .2/ sub  W  Z D sub  Z

D

@2 @2 @2 @2 @2  2  .x 2 C y 2 C 1/ 2  .w2 C x 2 C 1/ 2 C 2y 2 @x @y @w @z @x@w

C 2w

@2 @2 @2  2.yw C x/  2x : @x@z @w@z @y@w

When the aforementioned vector fields are considered on the group realized as a subgroup of the 4  4 matrix space, we have the following expressions: @ XQ D ; @x @ @ x2 @ YQ D Cx C ; @y @w 2 @z @ @ Cx ; WQ D @w @z @ ZQ D ; @z

9.2 Laplacian and Sub-Laplacian on Nilpotent Lie Groups

223

and .2/ sub D 

@2 @2 x 4 @2 @2  2  x2 2  2 @x @y @w 4 @z2

@2 @2 @2  x2  x3 ; @y@w @y@z @w@z   @2 @2 @2 x 4 @2 D  2  2  .x 2 C 1/ 2  x 2 C @x @y @w 4 @z2  2x

.1/

sub

@2 @2 @2  x2  .x 3 C 2x/ ; @y@w @y@z @w@z   @2 @2 @2 x 4 @2 2 2  D  2  2  .x C 1/ 2  1 C x C @x @y @w 4 @z2  2x

 2x

@2 @2 @2  x2  .x 3 C 2x/ : @y@w @y@z @w@z

More details about Engel’s group and its sub-Riemannian structure and a description of bicharacteristic curves of three-step Grushin operators can be found in Chap. 12 of the book [27] and in the paper [47].

Chapter 10

Heat Kernels for Laplacians and Step-2 Sub-Laplacians

10.1 Spectral Decomposition and Heat Kernel In this section we shall provide an explicit multiplication form of the Laplacian for a certain class of two-step nilpotent Lie groups including Heisenberg groups; see [46]. The result presented in this section is more precise than the explicit formula of the heat kernel given for the first time in the paper of Hulanicki [68], as explained in Remark 9.1.5. Let g be a two-step nilpotent Lie algebra such that g D gC ˚ g ˚ z; ŒgC ; g  D z; Œg˙ ; g˙  D 0

(10.1.1)

z is the center and n D dim gC D dim g ; dim z D d: Let fXi gniD1 , fYi gniD1 and fZk gdkD1 be a basis of gC , g and z, respectively, with the k : structure constants Ci;j d X k ŒXi ; Yj  D Ci;j Zk kD1

and all other brackets zero. For each  2 z , associate the matrix C./, C./i;j D

d X

k Ci;j .Zk /:

(10.1.2)

kD1

Assume the matrix C./ t C./ is diagonal: 0

c1 ./ B 0 B B C./ t C./ D B B @  0

 

c2 ./

0

  

1 C C C C: C A

cn ./

O. Calin et al., Heat Kernels for Elliptic and Sub-elliptic Operators, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-4995-1 10, c Springer Science+Business Media, LLC 2011 

225

226

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

with all the diagonal elements ci ./ nondegenerate positive bilinear forms on z. We identify the Lie algebra g D gC ˚ g  ˚ z and the Lie group exp W gC  g  z Š G through the exponential map. We denote an element g in G by gD

X

xi Xi C

X

yi Yi C

X

zk Zk D .x; y; z/:

The multiplication group law is defined as 1 g  gQ D g C gQ C Œg; g: Q 2 Consider the left invariant vector fields fXQ i g, fYQj g and fZQ k g expressed by X @ ei D @  1 Cikj yj X ; @xi 2 @zk X @ ej D @ C 1 ; Cikj xi Y @yj 2 @zk ek D @ : Z @zk The sub-Riemannian structure H on G is defined by the sub-bundle spanned by ei ; Y e j gn the vector fields fX i;j D1 . We choose the metric on H such that the aforementioned left invariant vector fields are orthonormal at each point. This left invariant metric on H can be extended to the entire tangent space as a left invariant ej ; Z e k g are orthonormal at each point. ei ; Y Riemannian metric by assuming that fX The sub-Laplacian sub and the Laplacian  are given by the formulas sub D  D

X

ei 2 C Y ei 2 ; X

X @2 X X @2 @2 @2 k k C  Ci;j yj C Ci;j xi 2 2 @xi @zk @yj @zk @xi @yi 2 X X X 1 @ k1 k2 C Ci;j Ci;j yj1 yj2 1 2 2 @zk1 @zk2 i

C

k1 4ˆ @zk ; : i D1 j D1 j D1 i D1 k 0

10.1 Spectral Decomposition and Heat Kernel

227

We shall give two examples next of the Lie algebras satisfying the above assumptions. Example 10.1.1. The Heisenberg algebra of any dimension. Example 10.1.2. The Heisenberg-type algebra with the dimension of the center d D dim z D dimŒg; g D 0 .mod 4/: Let fZk gdkD1 be an orthonormal basis of the center z and consider T D J.Z1 / ı J.Z2 / ı    ı J.Zd /: Using (9.2.4), we have T 2 D Id; T ı J.Z/ D J.Z/ ı T;

8Z 6D 0 2 z:

Now consider the vector spaces VC D fX 2 nI T .X / D X g;

V D fX 2 nI T .X / D X g:

Then for any nonzero Z 2 z, J.Z/ maps VC into V , and viceversa. We also have ŒVC ; V  D 0. For the Laplacian associated with this group, we have an explicit spectral decomposition in a multiplication form; see [46]. Theorem 10.1.3. There exist a measure space .X; dm/, a positive function ' on X and a unitary operator U W L2 .G/ Š L2 .X; dm/ such that U1 ı M' ı U D : Using the explicit form of the function ' and of the measure dm given by (10.1.3) and (10.1.6), we obtain the following consequence. Corollary 10.1.4. The spectrum of the Laplacian  is Œ0; 1/, and it consists of all continuous spectrum. Even if we do not intend to provide a proof for this theorem, we shall describe the measure space .X; dm/, the unitary transformation U and the positive function ' on X that appear in this case. Let k D .k1 ; : : : ; kn /; ki 2 N; ki  0, be a multi-index and define the measure space n p Y .Xk ; dmk / D gC  z nf0g; ci ./dvd 

i D1

! (10.1.3)

228

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

and consider the direct union .X; dm/ D

a

.Xk ; dmk /:

(10.1.4)

k2Nn

Let F be the partial Fourier transform F W C01 .gC  g  z/ ! C 1 .gC  g  z /; Z .nCd /=2 .F f /.x; ; / D .2/ e i.C/f .x; y; z/dyd z;

iD

p 1

and consider the restriction map R W L2 .gC  g  z / ! L2 .gC  g  .z nf0g//: We also let ˚ W gC  gC  .z nf0g/ ! gC  g  .z nf0g/ be a diffeomorphism defined by ˚.v; w; / D .x; ; /; x D v  w; 1 < ; y > D  < ; Œv C w; y > 2  D ;

ŒD T .v C w/.y/; y 2 g ;

where T W gC ! g is a isomorphism for any  ¤ 0. Then the composition K = ˚  ı R ı F is a unitary transformation given by K D ˚  ı R ı F W L2 .gC  g  z/ ŒD L2 .G/ n Y p



! L2 gC  gC  .z nf0g/;

!

ci ./dv dw d :

i D1

Consider next the following first-order differential operators: Si D

p @  ci ./wi ; i D 1; : : : ; n; @wi

and let hi be the functions given by 1

hi .w; / D e  2

p

ci ./w2 i

:

For any fixed  6D 0 and each multi-index k D .k1 ; : : : ; kn /, let h.w; ; k/ denote the Hermite function of the variables w 2 gC : h.w; ; k/ D .Sk11 h1 /.w; /    .Sknn hn /.w; /:

10.2 Complex Hamilton–Jacobi Theory

229

Now for a function f 2 C01 .gC  gC  .z nf0g//, we define 1 E.f /.v; ; k/ D Nk ./

Z gC

f .v; w; /h.w; ; k/d w 2 C 1 .gC  .z nf0g//;

where Nk ./ is the L2 -norm of the function h.w; ; k/: Z 2

Nk ./ D

gC

jh.w; ; k/j2 d w1   d wn D 2jkj kŠ n=2 

n Y

ci ./

ki 1 2

:

i D1

Then the operator E is extended to a unitary transformation from L2 .gC  gC  .z nf0g// to L2 .X; dm/, and the unitary operator U is defined by U D E ı ˚ ı R ı F:

(10.1.5)

Finally, let the function ' on X be '.v; ; k/ D jj2 C

n X

p .2ki C 1/ ci ./;

.v; / 2 gC  .z nf0g/:

(10.1.6)

i D1

With this introduction, we get the explicit integral expression for the heat kernel by calculating the kernel distribution of the composition operator U1 ı e t M' ı U: The heat kernel is given by the following result. Theorem 10.1.5. K.t I g; g/ Q D K.t I x; y; z; x; Q y; Q zQ / Z p Q D .2/.nCd /=2 e 1

(10.1.7)

z

e

t jj2

p p p c ./ cosh t c ./ ci ./  4i  sinh t pc i ./ f.xi xQi /2 C.yi yQi /2 g i d: p e  2 sinh t c ./ i i D1 n Y

10.2 Complex Hamilton–Jacobi Theory Complex Hamilton–Jacobi theory is different from the classical theory by the fact that the boundary values of the Hamiltonian system allow complex values, and hence the bicharacteristics and the action are complex. The fact that complex Hamilton–Jacobi theory is more suited to the study of sub-elliptic operators rather than classical theory was first noticed by Beals, Gaveau and Greiner in the 1990s; see [13, 14, 16]. The interested reader can also consult Chap. 5 of the book [28].

230

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

10.2.1 Path Integrals and Integral Expression of a Heat Kernel It is known from statistical mechanics that the heat kernel Kt .x; y/ must be expressed as a path integral Z e St . / d. /; K.t; x; y/ D Pt .x;y/

hopefully with a suitable infinite-dimensional measure d. /, where Pt .x; y/ denotes the path space connecting x to y at a time t. The function St . / is called the classical action and is given by Z 1 t 2 j .s/j P ds: St . / D 2 0 By normalizing the time parameter with t D 1, the aforementioned integral can also be written as Z S1 .t / 1 e  2t d. t /; with t . / D . t/: t N P1 .x;y/ In the case of the Laplacian the aforementioned integral has the following asymptotic expansion: K.t; x; y/ 

d.x;y/2 1 e  2t u0 .x; y/.1 C O.t//: n=2 .2 t/

Here d.x; y/ denotes the Riemannian distance between the points x and y. However, for sub-Laplacians the small time asymptotic expansion is more complicated: see [16]. In particular, when the Riemannian metric is Euclidean, there is only one geodesic (a line segment) that connects the points x and y, and hence the path integral reduces to the function 2 1  jxyj 2t e ; (10.2.8) .2 t/n=2 P @2 which is just the heat kernel of the Laplacian  D  12 niD1 @x 2. i

There are also similar arguments in the Heisenberg group case that reduce the path integral formula to a certain formula which makes full mathematical sense, see [78]. The heat kernel of the sub-Laplacian on the three-dimensional Heisenberg group was first constructed by Hulanicki in the mid-1970s, who provided an explicit integral formula using a probabilistic argument. Several papers published later in the 1990s deal with similar problems involving the heat kernel for Laplacians and sub-Laplacians on nilpotent Lie groups; see [14–16, 48, 98]. There are also quite a few recent papers dealing with a similar subject and calculations; see [32, 33, 78].

10.2 Complex Hamilton–Jacobi Theory

231

Unlike in Riemannian geometry, in sub-Riemannian geometry there are many geodesics connecting two points even locally; see [27, 28]. Because of this specific behavior, the heat kernel for sub-Laplacians will necessarily have an integral expression. There are several ways to attack the problem of heat kernels for sub-Laplacians. Some methods involve Mehler’s formula (see [107]), which is an important tool in the construction of the heat kernel on Heisenberg group, as done by Hulanicki [68]. This formula was also used in the previous section of this chapter to calculate the integrals included in the formula U1 ı e t M' ı U. This formula provides the generating function of Hermite polynomials explicitly. Another method is the method of complex Hamilton–Jacobi theory originated by Beals, Gaveau and Greiner [14–16], where there is no need for the generating function formula of Hermite polynomials. The aforementioned authors obtained the same formula directly, using that the heat kernel itself can be seen as a generating function. Their method starts by assuming that the heat kernel kt .g/ has a certain integral expression [see (10.2.10)]. The physical significance of the formula is that the heat flows mostly along the geodesics starting from the identity element at time t D 0, where the density of heat is the Dirac’s ı-function. The total amount of heat at a point g at a time t, denoted by kt .g/, should be equal to the sum (integral) over a certain class of geodesics arriving at point g at a time t from somewhere. This class of geodesics is determined by solving the Hamiltonian system (bicharacteristics system) under an initial-boundary condition; i.e., we assume that the coordinates in g=Œg; g are zero at t D 0 and the endpoint g must be arbitrarily given in the space G. In the Euclidean case there is only one such geodesic arriving at the point g under this condition, so no integration is needed and we have the well-known formula (10.2.8). However, in the nilpotent (non-abelian) cases we must consider geodesics whose initial points are not the identity element. These will be parameterized by the dual space Œg; g for both the Laplacian and sub-Laplacian cases. The reason for which we need to consider such geodesics is that on our curved space (which is otherwise topologically Euclidean) the wavefront set of the ı-function produces influence into the direction Œg; g. However, it is not clear whether this interpretation suffices to study the construction of the heat kernel on nilpotent Lie groups of step 3 or higher step, under the assumption that it has a prescribed integral form. Inspired by the heat kernel formula provided in Theorem 10.1.5, we shall consider the following ansatz as one of the possibilities of constructing the heat kernel: Theorem 10.2.1 (Meta-theorem). We assume that the heat kernel of a general nilpotent Lie group has the following integral form, where f denotes the action function and W denotes the volume element: K.tI .x; z/; .x; Q zQ// D kt ..Zx; Q zQ/1  .x; z//; (10.2.9)   f .x;z; / 1 e  t W .x; z; /d; .x; z/ D g 2 g=Œg; g  Œg; g ; kt .x; z/ D N t  (10.2.10) with a specific order N . For two-step nilpotent groups, N D 12 dim g=Œg; g C dimŒg; g.

232

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

Consequently, the function f .g; / will include all the information of the real geodesics when t & 0, and the volume element W .g; / will reflect the weight of the energy of the geodesics arriving at the point g. We shall briefly present the content of the following few sections. In Sect. 10.2.2 we shall explain a method to construct a solution of a class of Hamilton–Jacobi equations that gives the action function f assumed to exist by the Meta-theorem. In Sect. 10.2.3 we shall assume that the action function f is a solution of the following generalized Hamilton–Jacobi equation: X

i

@f .g; / C H.gI rf / D f .g; /; @i

and we shall find an equation satisfied by the volume element W .g; /, called the generalized transport equation. In Sects. 10.2.4 and 10.2.5 we shall show that this general mechanism works well in the case of two-step nilpotent Lie groups. In this case we have heat kernels for both sub-Laplacians and Laplacians in the integral form stated by the Meta-theorem.

10.2.2 A Solution of a Hamilton–Jacobi Equation In this section we shall consider a class of Hamilton–Jacobi equations and construct solutions under a certain assumption. This assumption is satisfied in the case of the Hamiltonian being the principal symbol of (left or right invariant) sub-Laplacians or Laplacians on two-step nilpotent Lie groups. Let H.x; yI ; / be a polynomial of the variables .x; yI ; / 2 Rm  Rd  Rm  d R and total degree 2 with respect to the variables  2 Rm and  2 Rd : H.x; yI ; / D

X

ai j .x; y/i j C

X

bi k .x; y/i k C

X

ck ` .x; y/k ` ; (10.2.11)

where ai j .x; y/, bi k .x; y/ and ck ` .x; y/ are polynomials of the variables x 2 Rm and y 2 Rd with real coefficients. We may also the functionsp ai j ; bi k and p allow 0 m ck ` to be entire functions of the variables x C 1x 2 C and y C 1y 0 2 Cd . This condition for the Hamiltonian is satisfied by the principal symbol of the (left)invariant sub-Laplacians and Laplacians on nilpotent Lie groups. Consider the Hamiltonian system xP D H D

@H.x; yI ; / ; @

@H.x; yI ; / ; P D Hx D  @x

yP D H D

@H.x; yI ; / ; @

P D Hy D 

@H.x; yI ; / ; @y

10.2 Complex Hamilton–Jacobi Theory

233

with the initial-boundary conditions 8 < x.0/ D 0; x.s/ D x; : .0/ D :

y.s/ D y;

The following assumption will play an important role in the sequel. Assumption 10.2.2 We assume that there exists an open conic domain D in Cd such that for any s 2 R, .x; y/ 2 Rm  Rd and  2 D, there exists a unique global solution of the above system X.t/ D X.tI s; x; y; /; .t/ D .tI s; x; y; /;

Y .t/ D Y .tI s; x; y; /; .t/ D .tI s; x; y; /;

which is smooth with respect to the variables .s; x; y; / 2 R  Rm  Rd  D. Under this assumption, let g be a function defined by the formula g.x; yI s;  / D

d X

j Yj .0I s; x; y;  /

j D1

C

Z sX

i .t/XPi .t/ C

X

j .t/YPj .t/  H.X.t/; Y.t/I .t/; .t//dt:

0

Then the function g D g.x; yI s; /, where the variable  is considered a parameter, is smooth and satisfies the following Hamilton–Jacobi equation: Proposition 10.2.3. We have @g C H.x; yI rg/ D 0; @s where rg D



@g @g @x ; @y

(10.2.12)

 .

Proof. This is proved by an explicit computation of the derivatives @g .x; yI s; /; @s

@g .x; yI s; /; @x

@g .x; yI s; /: @y

First we shall show that @g .x; yI s; / C H.x; yI .sI s; x; y; /; .sI s; x; y; // D 0: @s

(10.2.13)

234

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

We warn the reader that this will be a tedious computation: @g .x; yI s;  / @s d X @Yj j .0I s; x; y;  / D @s j D1 X X C i .s/XP i .s/ C .s/j YPj .s/  H.x; yI .sI s; x; y;  /; .sI s; x; y;  // Z s X @i .tI s; x; y;  /XPi .tI s; x; y;  / C @s 0 X @XPi .tI s; x; y;  / C i .tI s; x; y;  / @s X @j X @YPj C j .tI s; x; y;  / .tI s; x; y;  /YPj .tI s; x; y;  / C .tI s; x; y;  / @s @s X @H @Xi .tI s; x; y;  / .X.t/; Y.t/I .t/; .t//  @xi @s X @H @Yj  .tI s; x; y;  / .X.t/; Y.t/I .t/; .t// @yj @s X @H @i .X.t/; Y.t/I .t/; .t//  .tI s; x; y;  / @i @s  X @H @j .tI s; x; y;  / dt .X.t/; Y.t/I .t/; .t//  @j @s D

D

d X

@Yj .0I s; x; y;  / @s j D1 X X .s/j YPj .s/  H.x; yI .sI s; x; y;  /; .sI s; x; y;  // C i .s/XP i .s/ C ! ( Z s X Pi @i @ X .tI s; x; y;  /XPi .tI s; x; y;  / C i .tI s; x; y;  / .tI s; x; y;  / C @s @s 0 ! X @j Pj @ Y C .tI s; x; y;  /YPj .tI s; x; y;  / C j .tI s; x; y;  / .tI s; x; y;  / @s @s X X @Yj @Xi C Pi .tI s; x; y;  / .tI s; x; y;  / C P j .tI s; x; y;  / .tI s; x; y;  / @s @s  X X @j @i YPj .tI s; x; y;  / .tI s; x; y;  /  .tI s; x; y;  / dt XP i .tI s; x; y;  /  @s @s d X

j

@Yj .0I s; x; y;  / @s j D1 X X C i .s/XP i .s/ C .s/j YPj .s/  H.x; yI .sI s; x; y;  /; .sI s; x; y;  // ˇs X ˇs X @Yj @Xi ˇ ˇ .tI s; x; y;  /ˇ C .tI s; x; y;  /ˇ : j .t/ C i .t/ 0 0 @s @s j

10.2 Complex Hamilton–Jacobi Theory

235

Now from the initial-boundary conditions X.0I s; x; y; / D 0;

X.sI s; x; y; / D x;

Y .sI s; x; y; / D y;

we have @Xi XP i .sI s; x; y; / C .sI s; x; y; / D 0; @s @Yj .sI s; x; y; / D 0; YPj .sI s; x; y; / C @s @Xi .0I s; x; y; / D 0; @s which finally imply (10.2.13). Next we shall show that @g .x; yI s; / D i .sI s; x; y; /; @xk @g .x; yI s; / D l .sI s; x; y; /: @yl Another technical computation follows: @g .x; yI s; / @xk D

d X

j

j D1

Z

s

C 0

C

@Yj .0I s; x; y; / @xk X @i X @XPi .t I s; x; y; /XPi .t I s; x; y; / C i .t I s; x; y; / .t I s; x; y; / @xk @xk

X @j @xk X @H

.t I s; x; y; /YPj .t I s; x; y; / C

X

j .t I s; x; y; /

@Xi .X.t /; Y.t /I .t /; .t // .t I s; x; y; / @xi @xk X @H @Yj  .X.t /; Y.t /I .t /; .t // .t I s; x; y; / @yj @xk X @H @i .X.t /; Y.t /I .t /; .t // .t I s; x; y; /  @i @xk ! X @H @j  .X.t /; Y.t /I .t /; .t // .t I s; x; y; / dt @j @xk 

D

d X j D1

j

@Yj .0I s; x; y; / @xk

@YPj .t I s; x; y; / @xk

236

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

Z

s

C 0

X @i X @XPi .t I s; x; y; /XPi .t I s; x; y; / C i .t I s; x; y; / .t I s; x; y; / @xk @xk

X @j

X

@YPj .t I s; x; y; / @xk X X @Yj @Xi C Pi .t I s; x; y; / .t I s; x; y; / C Pj .t I s; x; y; / .t I s; x; y; / @xk @xk ! X X @ @ j i  XPi .t I s; x; y; / .t I s; x; y; /  YPj .t I s; x; y; / .t I s; x; y; / dt @xk @xk C

D

@xk

d X

j

j D1

.t I s; x; y; /YPj .t I s; x; y; / C

j .t I s; x; y; /

X @Yj @Yj ˇˇs @Xi ˇˇs X .0I s; x; y; / i .t I s; x; y; / j .t I s; x; y; / ˇ C ˇ @xk @xk 0 @xk 0

D k .sI s; x; y; /;

where we made use of the initial-boundary conditions @Xi .sI s; x; y; / D @xk @Yj .sI s; x; y; / D @xk

@xi D ıi;k ; @xk @yj D 0: @xk

@Xi .0I s; x; y; / D 0; @xk

Using the similar boundary conditions @Yj .sI s; x; y; / D @yl @Xi .sI s; x; y; / D @yl

@yj D ıj;l ; @yl @xi D 0; @yl

@Xi .0I s; x; y; / D 0; @yl

we have the following computation for the derivative of g: @g .x; yI s;  / @yl D

d X

j

j D1

Z

C 0

C

s

@Yj .0I s; x; y;  / @yl X X @i @XPi .tI s; x; y;  /XPi .tI s; x; y;  / C i .tI s; x; y;  / .tI s; x; y;  / @yl @yl

X @j @yl X @H

.tI s; x; y;  /YPj .tI s; x; y;  / C

X

j .tI s; x; y;  /

@Xi .X.t/; Y .t/I  .t/; .t// .tI s; x; y;  / @xi @yl X @H @Yj .X.t/; Y .t/I  .t/; .t// .tI s; x; y;  /  @yj @yl 

@YPj .tI s; x; y;  / @yl

10.2 Complex Hamilton–Jacobi Theory



X @H @i

.X.t/; Y .t/I  .t/; .t//

237

@i .tI s; x; y;  / @yl

! X @H @j .X.t/; Y .t/I  .t/; .t// .tI s; x; y;  / dt  @j @yl D

d X

j

j D1

Z

@Yj .0I s; x; y;  / @yl X @i X @XPi .tI s; x; y;  /XPi .tI s; x; y;  / C i .tI s; x; y;  / .tI s; x; y;  / @yl @yl

s

C 0

X @j

X

@YPj .tI s; x; y;  / @yl @yl X X @Yj @Xi .tI s; x; y;  / C P j .tI s; x; y;  / .tI s; x; y;  / C Pi .tI s; x; y;  / @yl @yl X @H @i  .X.t/; Y .t/I  .t/; .t// .tI s; x; y;  / @i @yl ! X @H @j .X.t/; Y .t/I  .t/; .t// .tI s; x; y;  / dt  @j @yl C

D

d X j D1

C

j

X

.tI s; x; y;  /YPj .tI s; x; y;  / C

j .tI s; x; y;  /

@Yj .0I s; x; y;  / @yl

i .tI s; x; y;  /

ˇs X ˇs @Yj @Xi ˇ ˇ .tI s; x; y;  /ˇ C j .tI s; x; y;  / .tI s; x; y;  /ˇ 0 0 @yl @yl

D l .sI s; x; y;  /:

Hence we conclude that g D g.x; yI s; / satisfies the equation @g C H.x; yI rg/ D 0: @s 

10.2.3 The Generalized Transport Equation In this section we shall deduce an equation, called the generalized transport equation, which is satisfied by the volume element present in the integral expression of the heat kernel for the sub-Laplacian sub on a nilpotent Lie group given by the Meta-theorem 10.2.1. Let G be an n-dimensional, connected, simply connected nilpotent Lie group with the Lie algebra g, and let fXi gm i D1 be a basis of the complement of the first derived ideal Œg; g; with m D dim g=Œg; g. Let m X sub D  XQ i2 (10.2.14) i D1

238

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

be the sum of the left invariant vector fields XQ i on the group G. Then sub is a sub-Laplacian satisfying the H¨ormander condition of hypoellipticity. To apply the result of Sect. 10.2.2, we fix a basis fZj gdjD1 of the derived ideal Œg; g, and we work with the coordinates G3 T  .G/ 3

X

ei C xi X

X

X

ei C xi X

ej $ .x; y/ 2 Rm  Rd ; yj Z

X

ej ; yj Z

X

i XQ i C

X

j ZQ j



$ .x; yI ; / 2 RmCd  RmCd ei ; Z e j g the dual basis of fXi ; Zj g. The corresponding where we have denoted by fX left invariant 1-forms are given by fi ; j g. This splitting .x; y/ of the coordinates corresponds to the splitting of the variables in the preceding section. Let H be the Hamiltonian associated with the sub-Laplacian (10.2.14): H W T  .G/ Š Rm  Rd  Rm  Rd 3 .x; yI ; / 7! H.x; yI ; / 2 R; 1X i .XQ i /2 : 2 m

H.x; yI ; / D

i D1

Assume the function f D f .x; yI / is a solution of the generalized Hamilton– Jacobi equation H.x; yI rf / C

d X i D1

i

@f D f .x; yI 1 ; : : : ; d /; @i

(10.2.15)

obtained by setting s D 1 in the solution g of the Hamilton–Jacobi equation (10.2.3) given in Sect. 10.2.2 under assumption (10.2.2). To make our assumption clear, we state once again that the heat kernel K.tI x; y; x; Q y/ Q is supposed to be of the form   Q y/ Q  .x; y/ ; K.tI x; y; x; Q y/ Q D kt .x; Z f .x;yI / 1 kt .x; y/ D N e  t W .x; y; /d; d t R

(10.2.16)

for some positive integer N > 0. In the case of a two-step nilpotent Lie group N is fixed and is given by N D 12 dim g=Œg; g C dimŒg; g D m=2 C d . However, the following calculations are valid for any d > 0 and N > 0. Let the characteristic variety of sub be Ch D f.x; y; ; /I H.x; y; ; / D 0g:

10.2 Complex Hamilton–Jacobi Theory

239

This is a sub-bundle in T  G and is trivialized as G  Œg; g ; which is embedded in T  .G/ according to the splitting g D Œg; g ˚ ŒfXi g: Ch D G  Œg; g T  .G/I see also Remark 10.2.12. The dimension of the integration variable  is d D dim Ch  dim G = dim Œg; g. In the following we shall accept the reasonable assumption that the integrand /  f .x;yW t e W .x; yI / decreases fast enough such that the partial integrations can be performed in a convenient way when jj ! 1; see also Theorem 10.2.7. Then we have sub .e 

f .x;yI / t

W/

 f X 1 1  ft H.x; yI rf /  W  e  .f /  W  X .f /X .W / e t  sub i i t2 t m

D

i D1

C sub .W /e

 ft

;

and @ @t

1 tN D

!

Z Rd

1 tN

e

 ft

Z Rd

 W d1    dd

1 N f  e t  W d1    dd C N t t

Z Rd

f f  e  t  W d1    dd : t2

Combining the last two relations, we obtain    @   sub C kt .x; y/ @t Z f 1 1 .H.x; yI rf /  f /  e  t  W d D N 2 t t Rd ! Z X f 1  XQ i .f /XQ i .W / C .sub .f /  N /  W e  t d t Rd i  Z f  sub .W /e  t d Rd Z X f 1 1 @f  e  t  W d D N  2 i t t Rd @i ! Z X f 1  XQ i .f /XQ i .W /  .sub.f / C N /  W e  t d t Rd i  Z f sub .W /e  t d  Rd  1 1 1  A2   A1  A0 : D N t t2 t

240

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

Now we assume W d D df ^ V

(10.2.17)

with the .d  1/-form1 V D

d X

b

.1/˛1 V˛ d1 ^    ^ d ˛ ^    ^ dd :

˛D1

In general, the coefficients Vi are functions of both space variables fxi ; yj g and the characteristic variety variables fj g. Therefore, the operation sub .V / is defined as sub .V / D

d X

b

.1/˛1 sub.V˛ /d1 ^    ^ d ˛ ^    ^ dd :

˛D1

The terms A1 and A2 will be computed under assumption (10.2.17) on the function W in the following way: 1  A2 t2

 Z X @f   f  1 e t df ^ V i D 2 t Rd @i   Z X  f @f 1 d e t ^ V i D t Rd @i X  Z f 1 @f D i e t d V t Rd @i 0 0 1 1 ! Z 2 XX X @f f 1 @ f D dV A e  t @df ^ V C @ i Vj A d C i t Rd @i @j @i j i i 0 1 Z Z 2 XX  f =t  1 @ f ^V  d e e f =t @ i Vj A d D t Rd @i @j Rd j i 0 1  X Z X @Vj @e f =t A d C i @ @i @j Rd j 0 1 Z 2 XX 1 @ f e f =t @ i Vj A d D t Rd @i @j j i 0 1 Z Z X X @2 Vj A d  .d C 1/ e f =t @ i e f =t dV:  d @ @ i j Rd R i j

1

P. Greiner worked out this form for the special case d D 1. See also Remark 10.2.4.

10.2 Complex Hamilton–Jacobi Theory

241

On the other hand, we have 1 1 A1 D t t

Z

X

Rd

Z

i

0

1 X @f XQ i .f /XQ i @ Vj A e f =t d @j j

 1  sub .f / C N e f =t df ^ V t d  Z RX  1 @ D H.x; yI rf /  Vj  e f =t d t Rd @j j Z Z XX     @e f =t Q Q d C sub .f / CN d e f =t ^ V  Xi .f /Xi .Vj / @j Rd j Rd i Z X @H.x; yI rf / 1 D e f =t Vj d t Rd @j j Z XX @   e f =t C XQi .f /XQ i .Vj / d @j Rd j i Z     e f =t d sub .f / C N V : 

Rd

Next we shall compute A0 C

1 A t2 2

 1t A1 . Using that for any j

X @ @2 f H.x; yI rf / C i D 0; @j @j @i i

we have 1 A1 t  Z X @f D sub Vj e f =t dt  .d C 1/ e f =t dV @j X X  Z @2 Vj f =t d i  e @i @j Z XX @    e f =t XQi .f /XQi .Vj / d @j Z C e f =t d ..sub .f / C N / V / ! Z Z X @f f =t D e sub Vj d C .N  d  1/ e f =t dV @j Z Z X X @2 Vj f =t i C e d.sub .f /V /  e f =t d @i @j  X X Z  @ Q Xi .f /XQ i .Vj / d  e f =t @j

A0 C

1 A2  t2 Z

242

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

  @f  X @f D  e f =t sub .Vj / d Vj C sub @j @j Z Z C .N  d  1/ e f =t dV C e f =t d.sub .f /V / X  Z Z X X @2 Vj X @Vj d  e f =t XQ i .f /XQ i i d  e f =t @i @j @j Z Z D  e f =t df ^ sub .V /  e f =t .sub .f /  N C d C 1/ dV X  Z Z X X @2 Vj X @Vj  e f =t i XQ i .f /XQ i d  e f =t d @i @j @j Z Z Z D  e f =t df ^ sub .V / C e f =t sub .f /dV C .N  d  1/ e f =t dV Z Z X XQ i .f /XQ i .dV/:  e f =t D.dV/  e f =t Z

Equating A0 C t12 A2  1t A1 D 0, it suffices for V to satisfy the following generalized transport equation: X XQ i .f /  XQ i .dV/ df ^ sub .V / C C D.dV/  .sub .f / C N  d  1/ dV D 0;

(10.2.18)

where D.V / is defined by D.V / D

X

i

b

XX @ @Vj .V / D .1/j 1 i d1 ^    ^ d j ^    ^ d @i @i

and we have D.dV/ D d D.V /  dV: If we assume that all the coefficients Vi depend only on the variables j , j D 1; : : : ; d , then the equation reduces to the transport equation for the two-step case. Remark 10.2.4. Fundamental solutions have volume elements which do not depend on the number of missing directions (see [13]). The result here is similar for the heat kernel volume element W , under the assumption that the volume element is of the form W d D df ^ V ; see (10.2.17). When there is only one missing direction, which, in the case of a nilpotent Lie group, means dimŒg; g D 1, we recover an unpublished result of P. Greiner. Next we shall work out the first-order transport equation. We consider    @  sub C kt .g/ @t ( Z f 1 1 .H.gI rf /  f /  e  t  W d D N 2 t t Rd

10.2 Complex Hamilton–Jacobi Theory

C

1 t Z

C D

1

Z

X

Rd

Rd

tN 1 C t Z C

i

Z

1 t2 Z

Rd

e  t d

)

X Rd

X

Rd

f

XQ i .f /XQ i .W /  .sub .f / C N /  W

sub.W /e

(

243

!

 ft

i

d

f @f  e  t  W d @i

!

XQ i .f /XQ i .W /  .sub .f / C N /  W

i

f

e  t d

)

sub.W /e

 ft

d

( Z X @  f 1 1 e  t  W d D N i t t Rd @i ! Z X f 1 XQ i .f /XQ i .W /  .sub .f / C N /  W e  t d C t Rd i ) Z  ft C sub.W /e d Rd

D

1

(

tN 1 C t Z C

1 t Z

D

(

tN 1 C t Z C

f

Rd

e t  X

Rd `

Rd

1

Z

i

Z

Rd

Rd

XQ i .f /XQ i .W /  .sub .f / C N /  W

f

Rd

 ft

e t 

X

! f

e  t d

)

sub.W /e

1 t Z

X @ .i W /d @i

d

X

i

@W d @i

XQ i .f /XQ i .W /  .sub .f / C N  d /  W

i

! f

e  t d

)

sub.W /e

 ft

d :

If the function W does not depend on the space variables, then W satisfies the firstorder transport equation X

i

@W  .sub .f / C N  d / W D 0: @i

(10.2.19)

244

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

10.2.4 Heat Kernel for the Sub-Laplacian This section deals with the complex Hamilton–Jacobi theory for two-step nilpotent Lie groups following [15]. In the sequel of this chapter we shall regard the heat equation as ! @ 1 C sub K D 0; @t 2 to avoid unnecessary constants in the description of the heat kernel. Let G be an .nCd /-dimensional connected, simply connected two-step nilpotent Lie group with center z D Œg; g of dimension dim z D d . We identify T  G with g  g . Let fXi gniD1 be a basis of the complement of the derived algebra Œg; g. Denote the coordinates on g  g by .x; zI ; / by fixing a basis fZj gdjD1 in the center z D Œg; g. d P k Let ŒXi ; Xj  D 2aij Zk , with aij D aj i , and let . / be a d  d matrix kD1

with the entries

. /i j D

d X

aikj k :

(10.2.20)

kD1

For each Xj we denote by XQ j D

@ @xj

C

d n P P i D1 kD1

k aij xi @z@k the corresponding left

invariant vector field on the group G. Then the sum sub D

n X

XQ i2

i D1

is a sub-Laplacian which satisfies the “H¨ormander condition” of hypoellipticity as before. The Hamiltonian associated with the aforementioned sub-Laplacian is given by d n n X X 1X aijk xi k H.x; zI ; / D j C 2 j D1

i D1 kD1

!2

X 1X D

. /j i  xi j  2 j

!2 :

i

We consider the Hamiltonian system 8 P ˆ xP j D Hj D j  . /j;i xi ; ˆ ˆ ˆ i < zP k D Hk ; ˆ ˆ Pj D Hxj ; ˆ ˆ : P D H 0; zk k

(10.2.21)

10.2 Complex Hamilton–Jacobi Theory

245

with the following initial-boundary conditions: 8 ˆ x.0/ D 0; ˆ ˆ ˆ ˆ < x.s/ D x D .x1 ; : : : ; xn / 2 Rn ; z.s/ D p z D .z1 ; : : : ; zd / 2 Rd ; ˆ ˆ ˆ .0/ D 1; ˆ ˆ :  D . ; : : : ;  / 2 Rd ; 1 d

(10.2.22)

where s 2 R, and x and z are arbitrarily given. Since X d 2 xj .t/ dxi .t/ D 2

. /j i ; 2 dt dt i we have x.t/ P D e 2t . / .0/. Then by integrating the equation

. /x.t/ P D . /e 2t . / .0/;  

. /x.t/ D 1=2 e 2t . /  Id .0/: p Now bypthe condition that the value .0/ D 1 is pure imaginary, the matrix 1 ./ is self-adjoint. Hence the matrix p Z 1 p 1 1s ./   d p D p   1s ./ sinh 1s ./ 2 1 sinh  we have

is well defined and invertible for any s 2 R and  2 Rd , so that we have a one-toone correspondence: between .0/ and x: p p 1 ./ s 1./  p  x.s/; s 6D 0: (10.2.23) .0/ D e sinh s 1 ./ The p contour  is taken to be suitably surrounding the spectrum of the matrix s 1 ./. Now we shall solve the following initial value problem: p P k p P 8 ˆ < xP j .t/ D Hj D j C 1 aij xi k D j  1 ./j i xi ; i i;k p P p P ˆ j  1 ./j ` x`  ./ij ; : Pi .t/ D Hxi D  1 j

`

with the initial conditions ( x.0/ D 0; .0/ D e s

(10.2.24)

p 1./



p 1./ p x: sinh s 1./

(10.2.25)

246

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

The solutions are given by x.t/ D x.tI s; x; / D e

p .st / 1./ sinh t

p

1 ./ p  x; sinh s 1 ./

.t/ D .t; s; x; / p   p p p 1 ./ p  e s 1./ Id  e t 1./ sinh t 1 ./  x D sinh s 1 ./ ! ! p p p p 1 ./ t 1./ s 1./ D e x cosh t 1 ./  e p sinh s 1 ./   p p D e t 1./ cosh t 1 ./ .0/: The solution x.t/ satisfies the boundary condition x.s/ D x; and this leads to the solutions of the initial-boundary value problem (10.2.21) under the condition (10.2.22), together with the solutions ! Z tX   X p 2u 1./ k .0/  ai j xi .u/ du; zk .t/ D zk C e s

.t/

j

j

p 1;

k D 1; : : : ; d;

i

 D .1 ; : : : ; d / 2 Rd :

It can be inferred from the previous expression that the functions zk .t/, with k D 1; : : : ; d; are uniquely determined, but since we do not need their explicit form in the following calculations, their final form won’t be provided. Let g D g.sI x; z; / 2 C 1 .R  Rn  Rd  Rd / be the complex action integral given by the formula d p X g.sI x; z; / D 1   zi .0I s; x; z; /

Z C

(10.2.26)

i D1 s

< .t/; x.t/ P > C < .t/; zP.t/ > H.x.t/; z.t/I .t/; .t//dt: 0

By Proposition 10.2.3, the action g satisfies the usual Hamilton–Jacobi equation @g C H.x; zI rg/ D 0: @s The function g also satisfies the relation g.sI x; z; `  / D

1  g.1I x; z; /: `

10.2 Complex Hamilton–Jacobi Theory

247

Next we shall provide an explicit determination of the function g.sI x; z; /. If we let p .t/ D x.t/ P D .t/  1 ./x; then

p P D 2 1 ./.t/; .t/

and we have g.sI x; z; / D

d p X 1 i  zi .0I s; x; z; / i D1

Z

s

C

< .t/; x.t/ P > C < .t/; zP.t/ > H.x.t/; z.t/I .t/; .t//dt 0

D

Z d p X 1 i  zi C p

1

d X

< .t/; x.t/ P > 0

i D1

D

s

Z

s

i  zi C

< .t/ C

p

1 < .t/; .t/ > dt 2

1 ./x;

0

i D1

1 < .t/; .t/ > dt 2 Z s d p p X 1 i  zi C < .t/; .t/ >  < x; 1 ./.t/ > dt D 1 0 2 .t/ > 

i D1

D D

p

1

p 1

d X

Z

s

i  zi C

1=2 < .t/; .t/ > C

i D1

0

d X

ˇs 1 ˇ i  zi C < x;  >ˇ 0 2

i D1

1 P > dt < x; .t/ 2

d E p X p  1 Dp D 1 i  zi C 1 ./ coth 1s ./  x; x : 2 i D1

Let f D f .x; z; / be given by f .x; z; / D g.1I x; z; / D

p

1

d X

i  zi C

i D1

E p 1 Dp 1 ./ coth. 1 .//  x; x : (10.2.27) 2

Then f satisfies the identity f .x; z; s  / D g.sI x; z; / s

248

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

and is a solution of the equation H.x; zI rf / C

d X

i

i D1

@f D f .x; z; 1 ; : : : ; d /; @i

called the generalized Hamilton–Jacobi equation. If  denotes the group law on the group G, then the heat kernel K.tI .x; z/; .x; Q zQ// D kt ..x; Q zQ/1  .x; z// is given by a function kt .x; z/ of the form Z f .x;z; / 1 e t  W .x; zI /d; d D dim Œg; g; kt .x; z/ D n=2Cd t Rd provided the function W .x; z; / satisfies the first-order transport equation (10.2.19):   X @W X d XQ j .f /XQ j .W /  sub .f / C i C W D 0: (10.2.28) @i 2 j

By noting that p  1 p tr 1 ./ coth 1 ./ 2 Z p 1  cosh   1   D  tr    1 ./ d 2 sinh 

sub .f / D

(10.2.29)

does not depend on the space variables .x; z/, we look for a solution of the transport equation (10.2.28) of the form W .x; z; / D W ./. In fact, the square root of the Jacobian of the correspondence (10.2.23) is a solution of the transport equation (10.2.28). Proposition 10.2.5. Let W ./ D det e

p 1./

p

1 ./ p  sinh 1 ./

! 12

! 12 p 1 ./ p D det ; sinh 1 ./

where the branch is taken to be W .0/ D 1. Then the function W ./ is a solution of the transport equation (10.2.28).   p 1t ./ p Proof. Let .t/ D W .t/2 D det . Then sinh

1t ./

  d X @W d .t/

.t/ D 2W .t/ k dt @k kD1 0 ! p d 1t

./ D tr @  p dt sinh 1t ./

p sinh

1t ./ p 1t ./

!1 1 A  .t/:

10.2 Complex Hamilton–Jacobi Theory

249

By making use of the resolvent equation, we have   Z 1  p d sinh    1 d  d

.t/  .t/ D tr     1t ./ dt d sinh   Z 1  p sinh    cosh  sinh      1t ./ D tr   d ;  sinh2  and then d X

@W ./ @k kD1 Z 1  p sinh    cosh  sinh   1   d  W ./:    1 ./ D tr 2  sinh2  k

Hence we have X @W ./  sub .f /  W ./ k @k Z 1  p sinh    cosh  sinh   1   d  W ./; D tr    1 ./ 2  sinh2   Z 1 p cosh   1  d  W ./ C tr    1 ./ 2 sinh  d D  W ./; 2 which shows that W ./ is a solution of the transport equation (10.2.28).



Remark 10.2.6. The function W ./ is similar to the van Vleck determinant (see [109] and [38]). We arrive at the following result. Theorem 10.2.7. The function kt .x; z/ is given by the following integral: 1 kt .x; z/ D .2 t/n=2Cd

Z Rd

e

/  f .x;z; t

!1=2 p 1 ./ p  det d; sinh 1 ./

where the action function f is given by (10.2.27): f .x; z; / D

d E p p X 1 Dp 1 i  zi C 1 ./ coth. 1 .//  x; x : 2 i D1

Proof. By the arguments of the last two sections, we know that ! @ 1 kt .x; z/ D 0; sub C 2 @t

250

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

since we have constructed the action function f and the volume form W in such a way that the function kt .x; z/ satisfies the heat equation. Using the following asymptotic behaviors,  W ./ D O.jjj /; j > 0 is arbitrary; D p E p   The bilinear form 1 ./ coth 1 .t/  x; x

(10.2.30)

is (strictly) positive definite and D p E p  1 ./ coth 1 .t/  x; x D O.jjjxj2 / (10.2.31) p (since nonzero eigenvalues of 1 ./ are proportional to jj ); we shall show that the Fourier inversion formula implies that 1 lim t !0 .2 t/n=2Cd

p

Z Z g

Rd

e

/  f .x;z; t

1 ./ p  det sinh 1 ./

!1=2 d'.x; z/dxdz (10.2.32)

D '.0; 0/ for 8' 2 C01 .g/. The computation will be provided in the following. Let F.z/ .'/.x; / D .2/d=2

Z

p

Rd

e

1

'.x; z/d z

be the partial Fourier transformation of functions '.x; z/ with respect to the variable z. Then !1=2 p 1 ./ p e  det d'.x; z/dxdz sinh 1 ./ g Rd Z Z 1 d=2 D F.z/ .'/.x; =t/ .2/ .2 t/n=2Cd !1=2 p p p 1 ./ ddx p  e 1=.2t /< 1./ coth 1./x;x> det sinh 1 ./ Z Z p 1 d=2 F.z/ .'/. tx; / .2/ D n=2Cd .2 t/ !1=2 p p p 1 .t/  e 1=2< 1.t / coth 1.t /x;x> det t d d t n=2 dx p sinh 1 .t/ Z Z <x;x> 1 t !0 F.z/ .'/.0; /e  2 ddx D '.0; 0/: ! n=2Cd=2 .2/ 

1 .2 t/n=2Cd

Z Z

/  f .x;z; t

10.2 Complex Hamilton–Jacobi Theory

251

10.2.5 Heat Kernel for Laplacian I In this section we shall provide the heat kernel for the Laplacian on the general two-step nilpotent Lie groups, based on our arguments about the results of the last section. Let  be a Laplacian on G  D sub 

d X

ZQ k2 ;

kD1

with ZQ k D @z@ . This means we are assuming that G is equipped with the left k invariant metric with respect to which fXi g and fZk g form an orthonormal basis at the identity element. The aforementioned operator becomes the Laplacian with respect to this Riemannian metric. Since we have Q D0 Œsub ; Z

.Z belongs to the center of g/;

the heat kernel K .tI .x; z/; .x; Q zQ// for the Laplacian  is the kernel distribution [which belongs to C 1 .RC  G  G/] of the composed operator  P 2 Q e t sub ı Id ˝e t Zk : Here Id denotes the identity operator on the space L2 .g=Œg; g/. We regard O 2 .Œg; g/ L2 .G/ Š L2 .g=Œg; g/˝L by the identification spanfXi g Š g=Œg; g:

Theorem 10.2.8. The heat kernel for the Laplacian on general two-step nilpotent Lie groups is given by Q zQ// K .tI .x; z/; .x; Z zj 2 1  jyQ 2t D K.tI .x; z/; .x; Q y//  e dy .2 t/d=2 center Z  Q z/1 .x;z/; / 1  f ..x;Q t e W ./d D .2 t/n=2Cd Rd Z f  .xx;zQ Q z1=2Œx;x; Q / 1 t e W ./d; D n=2Cd .2 t/ Rd

252

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

where f .x; z; / D

p

1 < ; z > C1=2
C1=2jj2: Remark 10.2.9. We shall show in the next section that the function f is the complex action integral for the Laplacian [see also (10.2.35)]. Proof. Since K .tI .x; z/; .x; Q zQ// is of the form K .tI .x; z/; .x; Q zQ// D kt ..x; Q zQ/1  .x; z// with function kt .x; z/ 2 C 1 .RC  g/, it will suffice to express this function as Z 2 1  jyj 2t dy kt .x; z/ D K.tI .x; z/; .0; y//  e .2 t/d=2 center Z Z p p p / coth 1. /x;x> 1  1C1=2< 1. t e W ./ D .2 t/n=2Cd Rd center jyj2 1 e  2t ddy  d=2 .2 t/ Z Z p 2 1 1 1  jyj 2t e t dy  e D n=2Cd d=2 d .2 t/ .2 t/ R center  e

p p p 1C1=2< 1. / coth 1. /x;x> t

Z

2  j2tj

W ./d

p p p / coth 1. /x;x>  1C1=2< 1. t

1 e e W ./d .2 t/n=2Cd Rd Z  / 1  f .x;z; t e W ./d: D  .2 t/n=2Cd Rd The aforementioned formula coincides with the earlier particular case of Theorem 10.1.5, and is the heat kernel we want to construct. D

10.2.6 Heat Kernel for Laplacian II In this section we shall construct the heat kernel for the Laplacian using the complex Hamilton–Jacobi method as in Sect. 10.2.4. We shall describe the Hamiltonian system and the related quantities associated with the sub-Laplacian. 1. The Hamiltonian H of the Laplacian: H .x; zI ; / 1 0 !2 d n n X d X X 1 k 1 @X j C a xi k C k2 A D 2 2 ij j D1

i D1 kD1

kD1

10.2 Complex Hamilton–Jacobi Theory

253

0

X 1 X j  D @

. /j i  xi 2 j

i

D H.x; zI ; / C

1 2

d X

!2 C

d X

1 k2 A

kD1

k2 :

kD1

2. The Hamiltonian system: 8 xP i ˆ ˆ ˆ < zP k P ˆ  ˆ j ˆ :P k

D H

D Hi D   . /x; i

D H  D Hk C k ; k D H

xj D Hxj ;

D H z 0:

(10.2.33)

3. The initial-boundary conditions: 8 ˆ x.0/ D 0; ˆ ˆ ˆ ˆ < x.s/ D x D .x1 ; : : : ; xn / 2 Rn ; z.s/ D zpD .z1 ; : : : ; zd / 2 Rd ; ˆ ˆ ˆ .0/ D 1; ˆ ˆ :  D . ; : : : ;  / 2 Rd ; 1 d

(10.2.34)

where s 2 R and x; z are arbitrarily given. In the Hamiltonian system (10.2.33), with the exception of the second equation, the other equations coincide with the corresponding equations in (10.2.21). Hence we have the same solutions x .t/ D x.t/ and  .t/ D .t/ given by x .t/ D x .tI s; x; / D x.tI s; x; / p p sinh t 1 ./ D e .st / 1./ p  x; sinh s 1 ./  .t/ D .t/ D .tI s; x; / p   p p p 1 ./ D p  e s 1./ Id  e t 1./ sinh t 1 ./  x sinh s 1 ./ ! p   p p p 1 ./ x p D e t 1./ cosh t 1 ./  e s 1./ sinh s 1 ./   p p D e t 1./ cosh t 1 ./ .0/;

254

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

and satisfy the system (10.2.24) under the initial conditions (10.2.25). The solution z .t/ is given by p z .t/ D z.t/ C 1.t  s/k : Now the complex action integral g 2 C 1 .R  Rn  Rd  Rd / is given by the formula g .sI x; z; / D

p

1

Z

d X

i  z

i .0I s; x; z; /

i D1 s

<  .t/; xP .t/ > C < .t/; zP .t/ >

C 0

 H .x .t/; z .t/I  .t/; .t//dt Z s d p X i  zi C < .t/; x.t/ P > D 1

(10.2.35)

0

i D1

 H.x.t/; z.t/I .t/; .t// C 1=2

d X

! k .t/

2

dt

kD1

D g.sI x; z; / C

d s X 2 k ; 2

(10.2.36)

kD1

where g D g.sI x; z/ is the complex action function constructed using (10.2.26). We conclude with the following result. Proposition 10.2.10. The following hold: 1. g satisfies the usual Hamilton–Jacobi equation @g

C H .x; zI rg / D 0: @s 2. The function g satisfies the relation g .sI x; z; `  / D

1

 g .1I x; z; /: `

We can see that the function g .1I x; z; / coincides with the function which appears in the integrand of the heat kernel given in the last section. By the same reason as in the case of the sub-Laplacian, the function f .x; z; / D g .1I x; z; / satisfies the generalized Hamilton–Jacobi equation. This can also be proved directly as in the following: H .x; zI rf /C

d X i D1

i

d d 2 X X @f

@f X 2 D H.x; zI rf /1=2 k2 C i C k @i @i i D1 kD1 kD1 X D f .x; zI / C 1=2 k2 D f .x; zI /:

10.2 Complex Hamilton–Jacobi Theory

255

The Laplacian .f / is given by  X  k2 D sub .f /: .f / D  f C 1=2 We also know that the volume form W ./ is the solution of the first-order transport equation (10.2.19):   X X @W d

Q

Q

Q Q i Xj .f /Xj .W / C Zk .f /Zk .W /  .f / C C W @i 2 j k   X @W d  W D 0: (10.2.37) D i  sub .f / C @i 2

X

Theorem 10.2.11. The heat kernel K for the Laplacian is given by Z 1  Q zQz 1 Q 2 Œx;x/ Q zQ/; .x; z// D e f .xx; W ./d: (10.2.38) K .tI .x; n=2Cd .2 t/ Rd Remark 10.2.12. This integral form coincides with the one obtained in Theorem 10.2.8. In both of the expressions given by Theorem 10.2.7 (sub-Laplacian case) and Theorem 10.2.11 (Laplacian case) the integrands of kt .g/ and kt .g/ are defined on G  Rd . We may identify this with the characteristic variety of the sub-Laplacian Ch D f.x; zI ; / j H.x; zI ; / D 0g through the following map: G  Rd 3 .x; z; / 7! .x; zI ./x; / 2 T  G: The characteristic variety is a sub-bundle in T  G and the integral formula of the heat kernel can be seen as the fiber integration of the d -forms f .x;z; / 1 e  t W ./d .2 t/n=2Cd

and

 / 1  f .x;z; t e W ./d .2 t/n=2Cd

on the characteristic variety Ch. On the other hand, for the special two-step case given by Theorem 10.1.3, the domain of integration in the expression of the heat kernel is the dual of the center Œg; g. It parameterizes the irreducible representation of G which appears in the  description of the unitary transformation U as L2 .G/ ! L2 .X/.

10.2.7 Examples In this section we shall express the heat kernels for the sub-Laplacian and Laplacian on the Heisenberg group, quaternionic Heisenberg group, free two-step nilpotent

256

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

Lie group of dimension 6, and Heisenberg-type groups in general. We only need to determine the explicit formula for the matrices p

p p 1 ./ coth 1 ./

and

1 ./ p det sinh 1 ./

!1=2 :

1. The three-dimensional Heisenberg group. Since the matrix . / for the threedimensional Heisenberg group is 

0 

 ; 0

we have   p p  coth  1 0 : 1 ./ coth 1 ./ D sinh  0 1 p   1 ./  1 0 p D sinh  0 1 sinh 1 ./ and then

!1=2 p 1 ./  D p det : sinh  sinh 1 ./

The action function f = f .x; y; z; / is f .x; y; z; / D

p  cosh  2 1z C .x C y 2 /: 2 sinh 

Then the heat kernel K.tI x; y; z; x; Q y; Q zQ/ for the sub-Laplacian sub D XQ 2  YQ 2 on the three-dimensional Heisenberg group is K.tI x; y; z; x; Q y; Q zQ/ (10.2.39)  Z C1 p1 zQzC yx Q xy Q  cosh  2 2 C 2 sinh  ..xx/ Q C.yy/ Q / 2  1 t e d: D 2 .2 t/ 1 sinh  Q y; Q zQ/ for the Laplacian  D XQ 2  YQ 2  ZQ 2 is The heat kernel K .tI x; y; z; x; given by K .tI x; y; z; x; Q y; Q zQ/ (10.2.40)  2 Z C1 p1 zQzC yx Q xy Q Q 2 C.yy/ Q 2 /  C  cosh  ..xx/ 2 2 sinh  2  1 t D d: e .2 t/2 1 sinh 

10.2 Complex Hamilton–Jacobi Theory

257

2. Higher-dimensional Heisenberg groups. Let H2nC1 be the .2n C 1/ -dimensional Heisenberg group described in Sect. 9.2.3 with the sub-Laplacian sub D 

X

XQ i2 

X

YQi2 :

In this case the matrix . /i j D ai j  is given by 

 0 Idn

. / D ; Idn 0 where Idn is the n  n identity matrix. We have the action function f expressed as f .x1 ; : : : ; xn ; y1 ; : : : ; yn ; z; / D

 p  cosh  X 2 1z C xi C yi2 ; 2 sinh 

and the volume element W ./ given by 

 n : sinh 

Hence the heat kernel K.tI x; y; z; x; Q y; Q zQ/ is given by the formula K.tI x; y; z; x; Q y; Q zQ/ Z C1 p1 1 D e .2 t/nC1 1   n  d: sinh 

P   P yQ i xi x Q i yi zQzC .xi x Q i /2 C.yi yQ i /2 C  cosh  2 2 sinh  t

. .

//

(10.2.41)

Q y; Q zQ/ for the Laplacian On the other side, the heat kernel K .tI x; y; z; x; D

X

XQ i2 

X

YQi2  ZQ 2

is given by K .tI x; y; z; x; Q y; Q zQ/ Z

D

C1

1 e .2 t/nC1 1   n  d: sinh 

P   p P P 2 yQ i xi x Q i yi C  cosh  .xi x Q i /2 C. yi yQ i /2   1 zQzC 2 2 sinh  2

.

/

t

(10.2.42)

258

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

3. The seven-dimensional quaternionic Heisenberg Lie group. Let qH7 (Š R7 ) be the seven-dimensional quaternionic Heisenberg group described in Sect. 9.2.4. e i , i D 0; 1; 2; 3, Recall the vector fields X X @ e0 D @ C 1 X xi ; @x0 2 @zi 3

i D1

e 1 D @ C x0 @ C x3 @  x2 X @x1 2 @z1 2 @z2 2 @ x x x @ @ 3 0 1 e2 D  C C X @x2 2 @z1 2 @z2 2 e 3 D @ C x2 @  x1 @ C x0 X @x3 2 @z1 2 @z2 2

@ ; @z3 @ ; @z3 @ : @z3

The sub-Laplacian and Laplacian in this case are given by sub D 

3 X

e i2 ; X

(10.2.43)

i D0

D

3 X

e i2  X

i D0

3 X @2 : @z2j j D1

(10.2.44)

Since the matrix . / has the expression 0

1 0 1 2 3 B 1 0 3  2 C C;

. / D B @ 2  3 0 1 A  3 2  1 0 we have

p

p 1 ./ coth . 1/ D jj coth jjId4 :

Hence the action function f is given by f .x; z; / D f .x0 ; x1 ; x2 ; x3 ; z1 ; z2 ; z3 ; 1 ; 2 ; 3 / D

p

1

3 X i D1

where jj D

qP

i2 and jxj D

qP

1 zi i C jj coth jj jxj2 ; 2

xi2 . The volume element in this case is 

W ./ D

jj sinh jj

2 :

10.2 Complex Hamilton–Jacobi Theory

259

Hence the heat kernel K.t; x; z; x; Q zQ/ for the sub-Laplacian (10.2.43) is given by the formula 1 K.t; x; z; x; Q zQ/ D .2 t/5

Z R3

e



p

1 t f



.x;Q Q z/.x;z/;/



 

jj sinh jj

2 d1 d 2 d 3 ;

  where f .x; Q Qz/  .x; z/;  is given by     p xQ 1 x0  xQ 0 x1 C xQ 3 x2  xQ 2 x3 f .x; Q Qz/  .x; z/;  D 1 1 z1  zQ1 C 2   xQ 2 x0  xQ 0 x2 C xQ 1 x3  xQ 3 x1 C 2 z2  zQ2 C 2   xQ 3 x0  xQ 0 x3 C xQ 2 x1  xQ 1 x2 C 3 z3  zQ 3 C 2 1 Q 2: C jj coth jj  jx  xj 2 The heat kernel of the Laplacian D

3 X

e 2i  X

i D0

3 X @2 @z2j j D1

is given by 1 K .t; x; z; x; Q zQ/D .2 t/5

Z e



p

1  t f



.x;Q Q z/.x;z/;

R3

  

jj sinh jj

2 d1 d 2 d 3 ;

  Q zQ; /C1=2 12 C22 C32 . where the action function f is given by f D f .x; z; x; 4. Free two-step nilpotent Lie group of dimension 6. This type of structure was introduced in Sect. 9.2.6. Let fg .3/ be the free two-step nilpotent Lie algebra of dimension 3 C C32 D 6 with an orthonormal basis fZi j g1i <j 3 of the center and an orthonormal basis fXi g3iD1 of the orthogonal complement of the center. The Lie bracket relations are ŒXi ; Xj  D ŒXj ; Xi  D 2Zi j ; for 1 i < j N; all others being zero. We shall express the heat kernel of the sub-Laplacian X XQ i2 ; sub D  and of the Laplacian  D sub 

X

ZQ i;j 2

260

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

where XQ i and ZQ i;j are the left invariant vector fields on the free two-step nilpotent Lie group fG .3/ Š fg .3/. It this case the matrix . / [see also (10.2.20)] is given as the most general 3  3 antisymmetric matrix 1 0 1 2

. / D @ 1 0 3 A ;  2  3 0 0

where D . 1 ; 2 ; 3 / 2 R3 is arbitrary. In the following we shall determine the volume element W ./ D det  p 1=2 p P 1./ p and the action function f .x; z; / D  k zk C 12 h 1 ./ sinh 1./ p p coth 1 ./ x; xi. Since q the eigenvalues of the self-adjoint matrix 1 ./ are f0; ˙jjg, with jj D 12 C 22 C 32 , we have !1=2 p 1 ./ jj det p D : sinh jj sinh 1 ./ p Let  be suitably chosen such that it is enclosing the eigenvalues of the matrix 1 ./. In order to calculate the expression Z

1 p

2 1



1 p  cosh   d;   1 ./ sinh 

it is enough to calculate the power series of the matrix p Id C

1 ./

2 C

2Š p

Id C

1 ./

2

3Š

C

p 4 1 ./ 4Š p 4 1 ./ 5Š

! C !1 C

:

For that purpose we consider the unitary matrix T : r

0 3 jj

B B B B B  2 T DB B jj B B B @ 1 jj

jj2 12 C22 32

12 C22 p 2.12 C22 /jj2 1jj1 C2 3

r

r

jj2 12 C22 32

p p 1jj2 1 3 2.12 C22 /jj2 1jj1 C2 3 jj2 12 C22 32

jj2 12 C22 32

2.12 C22 /jj2

r

1

 2 C22 p 1 2.12 C22 /jj2  1jj1 C2 3 C

r

2.12 C22 /jj2

r

jj2 12 C22 32

p jj2 12 C22 32  1jj2 1 3 p 2.12 C22 /jj2  1jj1 C2 3

C C C C C; C C C C A

10.2 Complex Hamilton–Jacobi Theory

which satisfies T

p 

261

0

1 0 0 0 1 ./T D @0 jj 0 A; 0 0 jj

for 1 6D 0. Then we have ! !1 1 p 1 p X X . 1 .//2k . 1 .//2k TT  Id C TT Id C .2k/Š .2k C 1/Š kD1 kD1 0 1 1 0 0 AT D T @0 jj coth jj 0 0 0 jj coth jj 0 2 1  2 3 1 3 1  jj coth jj @ 3 D jj coth jjId 3 C 2 3 22 1 2 A : jj2 1 3 1 2 12 

We can see that the resulting expression is also valid in the case 1 D 0. Now the action function is f .x1 ; x2 ; x3 ; z1 ; z2 ; z3 ; 1 ; 2 ; 3 / p X p p 1 D  1 k zk C < T T  1 ./ coth 1 ./ T T  .x/; x > 2 p X 1h k zk C jj coth jj.x12 C x22 C x32 / D  1 2 2 i 1  jj coth jj  C 3 x1  2 x2 C 1 x3 : 2 jj Hence the heat kernel K for the sub-Laplacian is given by the following expression: K.t; x; z; x; Q zQ/ 1 D .2 t/9=2 *0

Z R3

e

p  1

l1 .z1  Qz1 C 1=2.xQ2 x1  xQ1 x2 // C 2 .z2  Qz2 C 1=2.xQ3 x1  xQ1 x3 // C3 .z3  Qz3 C 1=2.xQ3 x2  xQ2 x3 // t

0

2

B 3 B B 1j j coth j j B B2 3 Bj j coth j jId3 C @ @ j j2

e

1 3

2t

2 3 22 1 2

11

+

1 3 CC CC C.xx/;x Q x Q 1 2 C AA 2 1

jj d1 d2 d3 : sinh jj

4. Heisenberg-type group. This structure was introduced in Sect. 9.2.5. Let V D n ˚ z be a Heisenberg-type algebra with the corresponding inner products <  ;  >n , <  ;  >z , and bilinear map J W z  n ! n, with the notations dim n D n and dim z D d .

262

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

The Lie bracket Œ ;  is defined by n  n ! z; .X; Y / 7! B.X; Y / D ŒX; Y ;

B.X; Y /.Z/ D < J.Z; X /; Y >n :

We shall use the same notation for the Lie algebra V and for the Lie group V . Let fXi g and fZ  k g be orthonormal bases on n and z , respectively. Denote by Q fXi g and fZQ k g the corresponding left invariant vector fields on the Lie group V , and consider the sub-Laplacian X

sub D  and the Laplacian

 D sub 

XQ i2 ;

X

ZQ k2 : d P

If the structure constants are given by ŒXi ; Xj  D 2

kD1

X

J.Z  ; / D

Ci kj Z  k , then

! D . /;

2Ci kj k

with

Z D

X

k Z  k 2 z :

k

Hence we have . /2 D j j2 Id . Now we can determine the action function f and the volume element W as in the following: p X p 1 p f .x; z; / D  1 k zk C h 1 ./ coth 1 ./x; xi 2 p X 1 k zk C jj coth jj  kxk2 ; D  1 2 s W ./ D

p



1 ./ D p sinh 1 ./

jj sinh jj

n=2 :

P P With x D xi Xi ; z D zi Z  i , the heat kernel K.t; x; z; x; Q zQ/ for the subLaplacian sub is given by 1 .2 t/n=2Cd

Z Rd

e

p  1

P



Q i zi Qzi B.x;x/ t





e



j jcoth j j jxxj Q

2 

2t

jj sinh jj

n=2 d z:

And the heat kernel for the Laplacian  is expressed as 1 .2 t/n=2Cd

Z Rd

e

p  1

P



i zi Qzi B.x;x/ Q t



2

e

 j2tj 



j jcoth j j jxxj Q 2t

2 

jj sinh jj

n=2 d z:

10.3 Grushin-Type Operators and the Heat Kernel

263

10.3 Grushin-Type Operators and the Heat Kernel In this section we shall introduce the Grushin-type operator associated with a sub-Riemannian structure and a submersion. This operator can be seen as a generalization of the Grushin operator 2 @2 2 @ C x : @x 2 @y 2

The heat kernel for such operators will be constructed in terms of the fiber integration of the heat kernel of a sub-Laplacian.

10.3.1 Grushin-Type Operators Let M be a manifold. We start by recalling the definition of a sub-Riemannian structure in a strong sense on M , that is, a sub-bundle H in the tangent bundle T .M / such that H is trivial as a vector bundle and all linear combinations of vector fields taking values in H and linear sums of finite numbers of their brackets span the entire tangent space T .M / at each point (such a sub-bundle is called nonholonomic). Denote the space of vector fields on M taking values in H by .M; H/. Usually this sub-bundle is equipped with a metric, and we call a manifold with such a sub-bundle a sub-Riemannian manifold in the strong sense. Let M be a sub-Riemannian manifold in the strong sense and let ' W M ! N be a (surjective) submersion. To define an operator on the manifold N in relation with H the sub-Laplacian on M , we assume that not only there are vector fields fXQ i gdim i D1 that trivialize the nonholonomic sub-bundle H, but also each Xi can be descended to the manifold N by the map '. This means: (Sub-1) The vector fields fXQ i g are linearly independent at each point on M and span the non-holonomic sub-bundle H. So they satisfy H¨ormander’s condition (bracket generating property; see [35] and [65]). Here we also consider a metric on H in such a way that the vector fields fXQ i g are orthonormal at each point on M . (Sub-2) We also assume that there is a volume form dM V on M with respect to which each vector field XQ i (i D 1; : : : ; dim H) is skew-symmetric. (Gru) If '.x/ D '.x 0 /, then d'x .XQ i / D d'x 0 .XQ i / for i D 1; : : : ; dim H. Let sub be the sub-Laplacian on M ; i.e., sub D 

dim XH

XQ k2 :

kD1

By the assumptions (Sub-1) and (Sub-2), sub is formally symmetric and hypoelliptic.

264

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

Definition 10.3.1. An operator G on N of the type GD

dim XH

2 d'.XQ k /

kD1

is called a Grushin-type operator. Note that the commutators ŒXQ j ; XQ k  can also be descended to N by the map ', so that we have the following results. Proposition 10.3.2. The operator G is hypoelliptic. Proposition 10.3.3. If each fiber of the map ' is compact, then G is formally symmetric with respect to the volume form ' .dM V / obtained by taking the push forward of the volume form dM V . Proof. Let f and g be in C01 .N /. We have Z

Z G .f /.y/  g.y/  .'/ .dM V /.y/ D '  .G.f //.x/  '  .g/.x/ dM V .x/ N M Z sub .'  .f //.x/  '  .g/.x/ dM V .x/ D M Z D '  .f /.x/  sub .'  .g//.x/ dM V .x/ M Z f .y/  G.g/.y/  .'/ .dM V /.y/: D  N

e k g and Remark 10.3.4. The sub-Laplacian depends on the specific vector fields fX is not uniquely defined as it is in the case of the Laplacian on Riemannian manifolds. In concrete cases these vector fields will be defined in a geometric way and will have a certain meaning for each case. If the fibers of the map ' W M ! N are not compact, we will have a volume form dN V on N such that '  .dN V / ^ D dM V , with a differential form on M of degree dim M  dim N . In our case the Grushin-type operators will be symmetric with respect to such a volume form. Similar considerations will apply to the Heisenberg group, the Engel group and the sphere. In the following we shall assume that the dimension dim H D dim N D n, Q are not linearly and let S be the submanifold on which T the vector fields fd'.Xk /g independent. This means that Hx Ker.d'x / 6D f0g for x 2 ' 1 .S/. So on the subset N nS the vector fields fd'.XQ i /g are linearly independent and span the tangent spaces there. We consider a Riemannian metric on the subset N nS in such a way that fd'.XQ k /g are orthonormal at each point of N nS: We call the manifold N with this Riemannian metric and the Grushin-type operator descended from a sub-Laplacian a singular Riemannian manifold (see [3]).

10.3 Grushin-Type Operators and the Heat Kernel

265

10.3.2 Heisenberg Group Case We shall show that the original Grushin operator is obtained in the way explained in the last section. Consider the three-dimensional Heisenberg group H3 , which is identified with R3 as a manifold, together with the following noncommutative group law:   .x; y; z/  .x; Q y; Q zQ/ D x C x; Q y C y; Q z C zQ C 1=2.x yQ  yx/ Q 2 R3 : The left invariant vector fields XQ and YQ are given by @  XQ D @x @ YQ D C @y

y @ ; 2 @z x @ : 2 @z

Q where ZQ D @ , the sub-bundle spanned by XQ Then from the relation ŒXQ ; YQ  D Z, @z and YQ is nonholonomic and defines a sub-Riemannian structure on H3 together with the metric Q YQ > D 0; < X; < XQ ; XQ > D < YQ ; YQ > D 1: The sum .XQ 2 C YQ 2 / is the sub-Laplacian sub and is symmetric with respect to the volume form dx ^ dy ^ d z, which coincides with the volume form introduced in Sect. 9.1.3. This form is the Haar measure of the group. We take a subgroup NY D f.0; y; 0/jy 2 Rg and consider the left quotient space Y W H3 ! NY nH3 , which is realized as Y W H3 Š R3 ! NY nH3 Š R2 ;  xy  Y .x; y; z/ D .u; v/ D x; z C : 2 The left invariance of the vector fields XQ and YQ satisfies condition (Gru) and is @ descended by the projection map Y to the vector fields d Y .XQ / D @u and @ Q d Y .Y / D u @v , respectively. The resulting operator  GD

2 @2 2 @ C u @u2 @v2



is the original Grushin operator. It is clear that this operator is symmetric with respect to the volume form du ^ d v and   .du ^ d v/ ^ dy = dx ^ dy ^ d z, where dy is a left NY -invariant 1-form (see Remark 10.3.4). As is easy to see, the vector fields d Y .XQ / and d Y .YQ / are linearly dependent along the line u D 0. We can consider a singular metric on the plane R2 in such a way that the two tangent vectors d Y .XQ /.u;v/ and d Y .YQ /.u;v/ are orthonormal at each point except on the line u D 0. The plane endowed with this (singular) metric will be called the Grushin plane.

266

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

If we consider the subgroup NZ D f.0; 0; z/jz 2 Rg, which is the center of the Heisenberg group, then the vector fields XQ and YQ can be descended to the quotient space H3 =NZ and the resulting vector fields on Z W R3 Š H3 ! H3 =NZ Š R2 ;

.x; y; z/ 7! Z .x; y; z/ D .u; v/ D .x; y/;

are dZ .XQ / D

@ @u

and

d Z .YQ / D

@ : @v

So in this case the Grushin-type operator is just the Euclidean Laplacian and it does not degenerate at any point. More generally, we take a subgroup N.a;b;c/ of the form N.a;b;c/ D f.at; bt; ct/ j t 2 Rg; where a2 C b 2 6D 0. We shall investigate only the case a 6D 0. The other case can be treated in a similar way. Let 0 W H3 ! N.a;b;c/ nH3 Š R2 . Then the map 0 is expressed as x H3 Š R 3 .x; y; z/ 7! .u; v/; u D ay  bx; v D z  a 3

  ay  bx cC : 2

We have a trivialization of the principal bundle 0 W H3 ! N.a;b;c/ nH3 given by R  R2 3 . I u; v/ ! .x; y; z/ 2 H3 ; where

u C b ; z D v C .c C u=2/: a We also have the coordinate expression of the cotangent bundle T  .H3 / as x D a; y D

'0

R  R3  R  R3 Š T  .H3 / Š R  R3  R  R3 ; ! .x; y; zI ; ; /;   x ay  bx x D ; u D ay  bx; v D z  cC ; a a 2   x  ; ˇ D : D a C b C  c C .ay  bx/=2 ˛ D C a 2a . ; u; vI ; ˛; ˇ/

(10.3.45) (10.3.46) (10.3.47)

The vector fields XQ and YQ are descended to the quotient space N.a;b;c/ nH3 , and the resulting vector fields are given by d0 .XQ / D b

cCu @ @  ; @u a @v

d0 .YQ / D a

@ : @u

10.3 Grushin-Type Operators and the Heat Kernel

267

It is easy to see that these vector fields are antisymmetric with respect to the volume form du ^ d v. With a left N.a;b;c/ -invariant 1-form dx on H3 , we have (see also a Remark 10.3.4) dx D dx ^ dy ^ dz: 0 .du ^ d v/ ^ a The Grushin-type operator in this case is GH

  @2 @ uCc @ 2 D b  a2 2 ; C @u a @v @u

(10.3.48)

and it does degenerate along the line f.u; v/ju D cg. Next we shall make a remark regarding the classification of one-dimensional subgroups of H3 . The group Aut.H3 / of the automorphisms of the Heisenberg group H3 is given by 8 19 0 0 = g11 g12 < Aut.H3 / D S D @g21 g22 0 A ; ; : a b det g   g11 g12 where g D 2 GL.2; R/ and .a; b/ 2 R2 are arbitrary vectors. For any g21 g22 subgroup N.a;b;c/ , with .a; b/ 6D 0, if we suitably choose an element S 2 Aut.H3 /, then S.N.a;b;c/ / is mapped to the subgroup N.0;1;0/ .

10.3.3 Heat Kernel of the Grushin Operator In this section we shall construct the heat kernel for Grushin operators coming from the sub-Laplacian sub on the three-dimensional Heisenberg group H3 by using the method of fiber integration. We shall use the same notations as in the last section. First, we shall construct the heat kernel of the original Grushin operator GD

@2 @2  u2 2 : 2 @u @v

Let K D K.t; x; y; z; x; Q y; Q zQ/ 2 C 1 .RC  H3  H3 / be the heat kernel of the sub-Laplacian sub D XQ 2  YQ 2 on the three-dimensional Heisenberg group H3 : 1 KD .2 t/2

Z

C1

1

  p Q xy Q  .xx/ Q 2 C.yy/ Q 2 1 t .zQzC yx /   coth 2t 2

e

e

 d: sinh 

268

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

Let KG du ^ d v be a 2-form on RC  H3  .NY nH3 / given as the fiber integration of the 3-form Kdx ^ dy ^ d z on RC  H3  H3 by the map Y W RC  H3  H3 ! RC  H3  .NY nH3 / Š RC  R3  R2 ;     xQ yQ I t; x; y; z; x; Q y; Q zQ 7! t; x; y; z; u; v ; u D x; Q v D zQ C 2 that is          KG t; x; y; z; u; v du ^ d v D Y  K t; x; y; z; : : : d xQ ^ d yQ ^ d zQ t; x; y; z; u; v ;



(10.3.49)



where the value at the point t; x; y; z; u; v 2 RC  H3  .NY nH3 / is given by the integration along the fiber Y1 .u; v/ for each fixed .t; x; y; z/ 2 RC  H3 . To do this integration, we shall describe the projection map Y as follows. Let a decomposition R  R2 of the group H3 be D W R  R2 Š R 3 Š H3 ;

 xs  .u; v; s/ 7! .x; y; z/ D u; s; v  : 2

Then we have Y ı D.u; v; s/ D .u; v/; and dx ^ dy ^ d z D du ^ ds ^ d v: Hence        Y  K t; x; y; z; : : : d xQ ^ d yQ ^ d zQ t; x; y; z; u; v "Z # C1   D K t; x; y; z; u; s; v  us=2 ds du ^ dv 1

D

1 .2 t/2

Z

C1

1

Z

C1

p

e

1 t .zvCus=2C sxuy / 2

1

  dds du ^ dv sinh  

Z C1 Z C1 p 1   2 1 2t s.xCu/   coth 2t s ds e e D .2 t/2 1 1     2 p    1 t zC xy v   coth xu 2 2t e d du ^ d v e sinh 

Z C1   p .uCx/2  1  zC xy v 1 1  8t t 2 coth  e e p D .2 t/3=2 1  coth    2  coth    2t xu d du ^ d v e sinh  e



   coth .xu/2 C.ys/2 2t



10.3 Grushin-Type Operators and the Heat Kernel

D

1 .2 t/3=2

Z

C1

e

  p .uCx/2 1 t zC xy  8t 2 v coth 

e

1

269

e



   coth xu 2t

2 r

  d du ^ d v cosh  sinh 

D KG .t; x; y; z; u; v/du ^ d v: Since the heat kernel K of the sub-Laplacian sub on H3 is invariant under the left action of the group H3 in the sense that K.t; g  .x; y; z/; g  .x; Q y; Q zQ// D K.t; x; y; z; x; Q y; Q zQ/; 8g 2 H3 ; and because of the commutativity of the left actions with the projection map previously can be seen as a smooth function on Y , the  function   KG obtained  RC  NY nH3  NY nH3 . Also, the commutativity sub ı .Y / D .Y / ı G says that the function KG is the heat kernel of the Grushin operator. This can be written as Theorem 10.3.5. The function KG satisfies

Z lim

t !0

 @ GC KG D 0; @t KG .t; uQ ; vQ ; u; v/f .u; v/dud v D f .Qu; vQ /; f 2 C01 .R2 /;

or, more generally, 8f 2 L2 .R2 ; dudv/: Next we shall consider the subgroup NZ D f.0; 0; z/ j z 2 Rg of H3 and the quotient space Z W H3 ! H3 =NZ Š R2 . In this case the resulting Grushin operator @2 @2 Z .XQ /2  Z .YQ /2 D  2  2 @u @v is just the Euclidean Laplacian and the heat kernel is well known in this case. This can also be deduced from the formula of the heat kernel on H3 by integrating along the fiber, as in the previous case. Then it suffices to calculate the integral Z

C1

K.t; x; y; z; x; Q y; Q zQ/d zQ 1

(10.3.50)

270

10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians

by applying the inversion formula of the Fourier transform Z

C1

1

  K t; x; y; z; x; Q y; Q zQ d zQ

Z C1 Z C1 p   Q xy Q  1  .xx/ Q 2 C.yy/ Q 2  1 t .zQzC yx /   coth 2t 2 dd zQ e e sinh  .2 t/2 1 1 Z   Z   p p Q xy Q  1  .xx/ Q 2 C.yy/ Q 2 1 t Qz  1 t .zC yx /   coth 2t 2 D e e e d d zQ sinh  .2 t/2 R R    Z   p p Q xy Q  zQ  1 .xx/ Q 2 C.yy/ Q 2 1  1 t .zC yx /   coth 2t 2 e  2 F e d zQ D sinh  t .2 t/2 R Q 2 C.yy/ Q 2 1  .xx/ 2t e D ; 2 t D

where F is the Fourier transform 1 F .f /. / D p 2

Z

p

e

1x

f .x/dx:

R

Remark 10.3.6. In the case of (10.3.49), the integrand of the heat kernel K p

e





 sinh   uy   2 p p   coth    coth  2 D e 1 2t s.xCu/ e  2t .ys/ e  1 t z 2 v e  2t xu

 .xu/2 C.ys/2 1 t .zvCus=2C sxuy /   coth 2t 2

e

 sinh 

can be integrated with respect to the variable s, keeping the other variables fixed. But in the case of (10.3.50), the order of integration of variables  and s cannot be changed. We shall next consider the subgroup N.a;b;c/ . We shall treat only the case a 6D 0, since the case b 6D 0 can be treated in a similar way. The heat kernel   K GH .t; u; v; uQ ; vQ / 2 C 1 RC  .N.a;b;c/ nH3 /  .N.a;b;c/ nH3 / of the Grushin-type operator GH provided by (10.3.48) is given by the integral 1 .2 t/2 e

Z Z e R

  p  1 t zv.cCu=2/C .u=aCb2/xay

R

 2 2   coth 2t ..xa / C.yu=ab / /

 dd : sinh 

Q and z D vQ C .c Q C uQ =2/, then this can be If we put x D auQ , y D .Qu=a C b / expressed in the form given in the following result.

10.3 Grushin-Type Operators and the Heat Kernel

Theorem 10.3.7. The heat kernel of the operator GH is K GH .t; uQ ; vQ ; u; v/   Z p 2 2 uu/  1  1 t vQv b.Q   .cC.Quu/ /  .Quu/ 2 coth a.a 2 Cb 2 / e 2t .a 2 Cb 2 / coth  2t .a Cb 2 / e D 3=2 .2 t/ R 1=2   d: .a2 C b 2 / sinh  cosh 

271

Chapter 11

Heat Kernel for the Sub-Laplacian on the Sphere S 3

11.1 Sub-Riemannian Structure and Sub-Laplacian on the Sphere S 3 This section deals with the study of the heat kernel of a sub-Laplacian on the three-dimensional sphere, and a Grushin-type operator on S 2 , called the spherical Grushin operator. This is a hypoelliptic operator on S 2 and is defined by the method explained in the previous chapter. The method of investigation is the explicit determination of eigenvalues and eigenfunctions of the Laplacian and sub-Laplacian in terms of harmonic polynomials (see [7–9, 97]).

11.1.1 S 3 in the Quaternion Number Field Let H be the quaternion number field over R, with the basis f1; i; j; kg specified in Sect. 9.2.4. We consider the three-dimensional sphere as the set of unit elements [Š Sp.1/] in H. Let h D x0 1 C x1 i C x2 j C x3 k 2 S 3  H and consider the curves fh  exp tigt 2R , fh  exp tjgt 2R and fh  exp tkgt 2R in S 3 . Then these define left invariant vector fields Xi , Xj and Xk on S 3 (in fact, defined on the whole H and tangent to S 3 ) and are expressed as follows: @ @ @ @ C x0 C x3  x2 ; @x0 @x1 @x2 @x3 @ @ @ @ Xj D x2  x3 C x0 C x1 ; @x0 @x1 @x2 @x3 @ @ @ @ C x2  x1 C x0 : Xk D x3 @x0 @x1 @x2 @x3 Xi D x1

It is easy to check that the previous vector fields are orthonormal at each point x 2 S 3 with respect to the Euclidean metric. Considered first-order differential

O. Calin et al., Heat Kernels for Elliptic and Sub-elliptic Operators, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-4995-1 11, c Springer Science+Business Media, LLC 2011 

273

11 Heat Kernel for the Sub-Laplacian on the Sphere S 3

274

operators, the vector fields Xi , Xj and Xk are skew-symmetric with respect to the following Riemannian volume form dS.x/ on S 3 : dS.x/ D x0 dx1 ^ dx2 ^ x3  x1 dx0 ^ dx2 ^ dx3 C x2 dx0 ^ dx1 ^ dx3  x3 dx0 ^ dx1 ^ dx2 :

(11.1.1)

The Lie algebra of the three-dimensional sphere S 3 , considered a Lie group Sp.1/, is generated by the basis i; j; k. The Lie brackets are defined to be the commutators of the elements. In particular, we have Œk; i D ki  ik D 2j and then ŒXi ; Xk  D 2Xj : Therefore, the sub-bundle H in T .S 3 / generated by the two vector fields fXi ; Xk g defines a minimal sub-Riemannian structure in the strong sense. Consequently, the operator sub on S 3 defined by sub D Xi2  Xk2

(11.1.2)

is the sub-Laplacian. Also, the operator  D Xi2  Xj2  Xk2 is the Laplacian on S 3 with respect to the standard Riemannian metric. Remark 11.1.1. The volume form constructed in Sect. 9.1.3 coincides with the Riemannian volume form (11.1.1) up to a multiplicative constant. Remark 11.1.2. The vector field N normal to S 3 is given by N WD x0

@ @ @ @ C x1 C x2 C x3 : @x0 @x1 @x2 @x3

We identify the set of quaternions H Š R4 with C2 by the following correspondence: (11.1.3) H 3 x D x0 1 C x1 i C x2 j C x3 k D .x0 1 C x2 j/ C i.x1 1 C x3 j/ p p   $ .x0 C x2 1; x1 C x3 1/ DW w0 .x/; w1 .x/ D .w0 ; w1 / 2 C2 ; and we consider the space H as a (right-)complex vector space with the complex multiplication p H  C 3 .x; / D .x0 1 C x1 i C x2 j C x3 k; a C 1b/ D .w0 ; w1 ; /   7! .w0 ; w1 / D x0 1 C x2 j C i.x1 1 C x3 j/  .a1 C bj/ 2 H:

11.1 Sub-Riemannian Structure and Sub-Laplacian on the Sphere S 3

275

Under this identification, the one-parameter transformation group generated by the vector field Xj is the scalar multiplication by  D a1 C bj 2 U.1/ from the right. The orbit space of this action on S 3 is the complex projective space P 1 C, and we have the following principal bundle, called the Hopf bundle: R W S 3 ! P 1 C: Let ` W U.1/ ! U.1/ be the character ` ./ WD ` . We denote the complex line bundle on P 1 C associated with the character ` by L` . Let PN Œ x0 ; x1 ; x2 ; x3  D PN Œ x  be the space of homogeneous polynomials of degree N in the variables x0 ; x1 ; x2 and x3 . We denote by HN D HN Œ x  the subspace of harmonic polynomials in PN Œ x : ( ) 3 ˇ X @2 p.x/ ˇ HN Œ x  D p 2 PN Œ x  ˇ D0 : (11.1.4) @xi2 i D0 Proposition 11.1.3. The following decomposition of PN Œ x  holds: P N Œ x  D HN Œ x  C

3 X

! xi2

 PN 2 Πx :

i D0 3 The space HN Πx  restricted  2 to the2 sphere2 S , denoted by HN , is the eigenspace of the Laplacian  D  Xi C Xj C Xk with respect to the eigenvalue N D N.N C 2/. Moreover, its dimension is given by

dim HN D .N C 1/2 D dim HN Πx : We shall express the elements of PN Πx  in terms of the complex and anticomplex variables w0 ; w1 ; w0 and w1 given by (11.1.3). We shall denote by P.n;m/ the subspace of PN Πx  = PN Πw0 ; w1 ; w0 ; w1  (with n C m D N ) defined by P.n;m/ D

n

ˇ ˇ p. w0 ; w1 ; w0 ; w1 / 2 PN Œ w0 ; w1 ; w0 ; w1  ˇ p p p p   p w0  e 1t ; w1  e 1t ; w0  e 1t ; w1  e 1t o p   D e 1.nm/t p w0 ; w1 ; w0 ; w1 ; 8 t 2 R :

Let N D n C m and H.n;m/ WD P.n;m/

T

HN . Then we have

HN Πx  D HN Πw0 ; w1 ; w0 ; w1  D

X nCmDN n0;m0

˚ H.n;m/ ;

11 Heat Kernel for the Sub-Laplacian on the Sphere S 3

276

and the following decomposition holds:   P.n;m/ D H.n;m/ C w0 w0 C w1 w1  P.n1;m1/ : For any p 2 H.n;m/ , the following relation holds:   p D  Xi2 C Xk2 p C .n  m/2 p D sub p C .n  m/2 p:

(11.1.5)

The restriction of H.n;m/ to S 3 is denoted by H.n;m/ . Relation (11.1.5) implies the following result. Proposition 11.1.4 ([7, 9]). The space H.n;m/ is the eigenspace of the subLaplacian sub corresponding to the eigenvalue N.N C 2/  `2 D 4m2 C 4m.1 C j`j/ C 2j`j; with multiplicity dim H.n;m/ D j`j C 2m C 1; where ` D n  m, N D n C m, n  0; m  0. Let F ` be the subspace of C 1 .S 3 / defined by n

ˇ   f 2 C 1 .S 3 / ˇ f x  .a C bj/ o p D .a C b 1/` f .x/; a2 C b 2 D 1; a; b 2 R :

F` D

Then F ` can be identified with the space of smooth sections .L` / of the complex line bundle L` . Due to this identification and since csub .F ` /  F ` , the sub-Laplacian sub can be seen as a second-order differential operator acting on .L` /:     D` W  L` !  L` : The operator D` is elliptic and is called the horizontal Laplacian; see [7]. In fact, the principal symbol .D` /.x; / W L` ! L` , where .x; / 2 T  .P 1 C/nf0g, is given by  2  2       .f //.x/ Q C Xk .R .f //.x/ Q Q  D` .x; / D Xi .R  .s/.x/; with s 2 .L` / and f 2 C 1 .P 1 C/, such that df .x/ D 2 Tx .P 1 C/; f .x/ D 0, and R .x/ Q D x. In other words, the principal symbol is given as the multiplication by the nonzero number 

 .f //.x/ Q Xi .R

2

 2  C Xk .R .f //.x/ Q :

11.1 Sub-Riemannian Structure and Sub-Laplacian on the Sphere S 3

277

Remark 11.1.5. For each ` 2 Z, the minimal eigenvalue of the horizontal Laplacian D` is 2j`j with multiplicity j`j C 1. Proposition 11.1.6. For fixed ` 2 Z, the sum X

˚ H.n;m/ D .L` / Š F `

mnD` n0;m0

is the eigenspace decomposition of .L` / with respect to the operator D` .

11.1.2 Heat Kernel of the Sub-Laplacian on S3 We start this section by recalling certain operators discussed in [97] (see also [8]) that are used to express the heat kernels of the horizontal Laplacian D` and the sub-Laplacian sub and also the Laplacian on S 3 . Define the map n o  

S W T S 3 nf0g D .x; y/ 2 R4 R4 j jxj D 1; < x; y >D 0; y 6D 0 ! C4 nf0g by p   T S 3 nf 0 g 3 .x; y/ 7! S .x; y/ D z D j y jx C 1y 2 C4 nf 0 g:   The image XS D S T .S 3 /nf 0 g is a quadric hypersurface 4 ˇ n o X ˇ zi2 D 0 : XS D z D .z0 ; z1 ; z2 ; z3 / 2 C4 nf0g ˇ z2 D i D0

We identify the tangent bundle T .S 3 / and the cotangent bundle T  .S 3 / through the standard Riemannian metric. Then under the isomorphism S , the symplectic form !S on T  .S 3 /nf0g is expressed as a K¨ahler form v 1 u 3 uX p p jzi j2 A : !S D S @ 2 1 @ @t 0

i D0

Let PN Œ z  D PN Œ z0 ; z1 ; z2 ; z3  denote the homogeneous polynomials of degree N in the Pcomplex variables .z0 ; z1 ; z2 ; z3 /. We fix an inner product < ;  >XS on the space ˚PN Œ z  of all polynomials restricted to the quadric hypersurface XS by N 0

Z < f; g >XS D

  f .z/g.z/ exp  „jzj  jzjq S ; XS

11 Heat Kernel for the Sub-Laplacian on the Sphere S 3

278

where „ > 0; q > 3 are fixed constants and S D 

1 !S ^ !S ^ !S 3Š

is the Liouville volume form on T  .S 3 /nf0g [we identify T  .S 3 /nf0g with XS via

S ]. With respect to this inner product, the spaces Pk Œz and Pl Œz (restricted to XS ) are orthogonal for k 6D l. On the space HN (the space of harmonic polynomials of degree N restricted to the sphere S 3 ), we consider the map AN W HN ! RPN Œ z ; ' 7! S 3 '.x/ < x; z >N dS.x/;

(11.1.6)

3 P where z 2 XS and < x; z > D xi zi is a bilinear form on R4  C4 . i D0 P 2 P @2 N N 2 zi for each fixed z 2 XS , the < x; z > D < x; z > Since 2 @x i

polynomial 'z .x/ D < x; z >N of the variable x is harmonic. Hence, if (k 6D N ), then Z S3

2 Hk

.x/ < x; z >N dS. x / D 0:

Next, we define a map BN by BN W PN Œ z  ! RHN ;   7! XS .z/ < x; z >N exp  „jzj  jzjq S :

(11.1.7)

Since the integration in (11.1.7) is taken over XS , it can easily be checked that BN . / 2 HN . Proposition 11.1.7 ([8,97]). With the identity operator Id on HN , the composition BN ı AN has the form BN ı AN D aN Id; where the real constant aN is given by .2N C q C 3/ N C1 2N  „ .N C 1/3 with 2 y 4Vol. S 3 /  Vol. S 3 p / .1= 2/ VW D : „3Cq aN D V 

(11.1.8)

Here we denote by Vol.S 3 / [respectively, Vol. S 3 p2/ /] the volume of the unit p.1= 3 p sphere S 3 [resp. the sphere S.1= of radius 1= 2]. 2/ From now on we shall work in the coordinates .w0 ; w1 / D .w0 .x/; w1 .x// described in (11.1.3) rather than using .x0 ; x1 ; x2 ; x3 / 2 R4 . We also define new

11.1 Sub-Riemannian Structure and Sub-Laplacian on the Sphere S 3

279

coordinates .u0 ; u1 ; v0 ; v1 / in C4 [instead of .z0 ; z1 ; z2 ; z3 / by the coordinate transformation p p zi  1zi C2 zi C 1zi C2 ui D and vi D ; i D 0; 1: 2 2 P Then the bilinear map R4  C4 ! C .x; z/ 7! xi zi D< x; z > is expressed as < x; z > D w0 u0 C w0 v0 C w1 u1 C w1 v1 ; and

3 P i D0

zi 2 D 0 holds if and only if u0 v0 C u1 v1 D 0. The Laplacian  on R4 in

coordinates .w0 ; w1 / is expressed in the form 4



 @2 @2 C : @w0 @w0 @w1 @w1

Therefore, with the notations (11.1.4), we have Lemma 11.1.8. ( HN Πx  D

1 ˇ X @2 f ˇ f 2 PN Œ w0 ; w1 ; w0 ; w1  ˇ D0 @wi @wi

) :

i D0

With .r; s/ 2 N0 N0 , let Ar;s D Ar;s .w0 ; w1 ; w0 ; w1 I u0 ; u1 ; v0 ; v1 / be the function defined by r  s  Ar;s .w0 ; w1 ; w0 ; w1 I u0 ; u1 ; v0 ; v1 / D w0 u0 C w1 u1 w0 v0 C w1 v1 : Considered a function of the variables w0 ; w1 and w0 ; w1 , for any fixed .u0 ; u1 ; v0 ; v1 / with u0 v0 C u1 v1 D 0, it can be checked that Ar;s is a harmonic polynomial of degree r C s. Let Ar;s be the operator Ar;s W

X

˚HN Œ x  ! f 7!

X Z

˚PN Œ u; v 

r  f .w0 ; w1 ; w0 ; w1 / w0 u0 C w1 u1 S3 s   w0 v0 C w1 v1 dS.x/:

Proposition 11.1.9. Using the notations of Sect. 11.1.1, we have (i) Ar;s . HN / D 0 if N 6D r C s (ii) Ar;s . H.k;l/ / D 0 for r C s D k C l, but r  s 6D k  l

11 Heat Kernel for the Sub-Laplacian on the Sphere S 3

280

˛

˛

ˇ

ˇ

0 1 0 1 The image P Ar;s . H.r;s/ P / is spanned by the monomials of the form u0  u1  v0  v1 with ˛i D r and ˇi D s.

Put Pr;s Πu; v  WD Ar;s . H.r;s/ /. Then it follows that Pr;s Πu; v  and Pk;l Πu; v  are orthogonal when r C s 6D k C l or r  s 6D k  l in case r C s D k C l. We conclude these properties with the following result. Proposition 11.1.10. The map AN in (11.1.6) can be decomposed in the form N X

CNr AN r;r D AN ;

CNr D

where

rD0

NŠ : .N  r/ŠrŠ

Moreover, we have Proposition 11.1.11. The inverse operator A1 r;s W Pr;s Πu; v  ! H.r;s/ is given by a constant Cr;s times the integral operator Br;s W Pr;s Πu; v  ! H.r;s/ :  1=2 , then Br;s is defined by If we put j.u; v/j WD ju0 j2 C ju1 j2 C jv0 j2 C jv1 j2 Br;s W

7! 2

qC3 2

Z .u; v/ XS

1 X

!r wi u i

i D0

Again, we have a decomposition

!s

p 2„ j.u;v/j

wi vi j.u; v/jq e 

S .u; v/:

i D0 N P

rD0

BN ı AN D



1 X

N X

CNr BN r;r D BN and

CNr 2 BN r;r ı AN r;r D aN Id:

rD0

Moreover, on each subspace H.N r;r/ we have BN r;r ı AN r;r D

aN . r /2 .CN

We shall express the heat kernel k ` .tI x; y/ D k ` .tI w.x/; w.y// of the operator sub acting on the space F ` in terms of the variables p   w.x/ D w0 .x/; w1 .x/ ; wi .x/ D xi C 1xi C2 ;

where i D 0; 1:

Since the vector fields Xi and Xk are left invariant, the kernel function k ` satisfies the invariance     k ` tI g  x; g  y D k ` tI x; y

with

g; x; y 2 S 3 :

(11.1.9)

The function space .L` / is left invariant as well. In the following proposition we give an expression of the kernel k ` .tI x; y/ (with x; y 2 S 3 ) of the operator e tD` acting on the complex line bundle L` .

11.1 Sub-Riemannian Structure and Sub-Laplacian on the Sphere S 3

281

Proposition 11.1.12. Let `  0 and x; y 2 S 3 . Then the heat kernel k ` .tI x; y/ can be expressed as   k ` tI x; y  m 2 1   qC3 X C 2 2mC` 2 e t 4m C4m.1C`/C2` D2 a2mC` mD0 Z X mC` X m X mC` X m  wi .x/ ui   wi .y/ vi wi .x/ vi  wi .y/ ui XS

p

j.u; v/jq  e  2„j.u;v/j S .u; v/  m 2 1   qC3 X C 2 2mC` e t 4m C4m.1C`/C2` D2 2 a2mC` mD0 Z X mC`  X m  wi .y 1 x/ ui   u0mC`  vm wi .y 1 x/ vi 0 XS

p

j.u; v/jq  e 

2„j.u;v/j

S .u; v/:

Note that w0 .y 1 x/ D w0 .y/w0 .x/ C w1 .y/w1 .x/; w1 .y 1 x/ D w0 .y/w1 .x/  w1 .y/w0 .x/: The kernel function k ` satisfies the identities p p     (i) k ` tI x  e 1 ; y D e  1 `  k ` tI x; y ; p p     (ii) k ` tI x; y  e 1 D e 1 `  k ` tI x; y . In the case ` < 0 one has   k ` t I x; y D2

qC3 2

 m 2 1 X C2mCj`j

mD0

Z

X

 XS

e t





4m2 C4m.1Cj`j/C2j`j

a2mCj`j  m X mCj`j  X m  X mCj`j wi .x/ ui  wi .x/ vi  wi .y/ ui  wi .y/ vi p

j.u; v/jq  e  2„j.u;v/j S .u; v/  m 2 1 C2mCj`j t 4m2 C4m.1Cj`j/C2j`j qC3 X e D2 2 a2mCj`j mD0 Z X m  X mCj`j mCj`j  wi .y 1 x/ ui  wi .y 1 x/ vi  um 0  v0 XS

p

j.u; v/jq  e 

2„j.u;v/j

S .u; v/:

11 Heat Kernel for the Sub-Laplacian on the Sphere S 3

282

For ` < 0, the kernel k ` satisfies equivalence properties similar to (i) and (ii) above. The heat kernel Ksub . tI x; y / of the sub-Laplacian can be obtained as the sum over the kernels k ` . tI x; y /, ` 2 Z. Theorem 11.1.13.   Ksub tI x; y D2

 m 2 1 1 X X C2mC`

qC3 2

Z 

e t





4m2 C4m.1C`/C2`

a2mC` mC` X m X mC`  X m wi .x/ui   wi .y/vi wi .x/vi  wi .y/ui

`D0 mD0

X XS

p

j.u; v/jq  e 

2„j.u;v/j

S .u; v/  2 1 1   m qC3 X X C 2 2mC` 2 C2 e t 4m C4m.1C`/C2` a2mC` `D1 mD0 Z X m X mC` X m X mC` wi .x/ui   wi .y/vi wi .x/vi wi .y/ui   XS

p

j.u; v/jq  e 

2„j.u;v/j

S .u; v/:

11.1.3 Spherical Grushin Operator and the Heat Kernel ˚ In this section we consider the left action of the group U.1/ Š e t j gt 2R on S 3 , and define a Grushin-type operator on P 1 .C/; i.e., we consider the Hopf bundle L W S 3 ! U.1/nS 3 Š P 1 C [= the orbit space by the left action of U.1/]. Then, by the associative law of the quaternion number field, the operators Xi and Xk commute with the left action and one obtains vector fields dL .Xi/ and dL .Xk / on P 1 C. Considered first-order operators, these vector fields are skew-symmetric with respect to the volume form .L / .dS.x// [the forward push of the volume form dS.x/ by the projection map L ; see Sect. 9.1.3]. This volume form also coincides with the standard volume form on S 2 except for a multiplicative constant. Since the vector fields Xi and Xk satisfy conditions (Sub-1), (Sub-2) and (Gru), we have Proposition 11.1.14. The operator GS WD dL .Xi /2  dL .Xk /2 is hypoelliptic on P 1 C. We shall call the operator GS the spherical Grushin operator, since it is con 2 @ @2 structed in a similar way as the Grushin operator GH WD  @x 2 C x 2 @y 2 .

11.1 Sub-Riemannian Structure and Sub-Laplacian on the Sphere S 3

283

Remark 11.1.15. All elements h 2 H with h2 D 1 are conjugate to each other. Therefore, taking another generator g0 (g0 2 D 1) of the group U.1/, the resulting operator on P 1 C is conjugate to GS as well. We determine the subset in P 1 C on which the operator GS degenerates, i.e., the subset on which the vector fields dL .Xi / and dL .Xk / are linearly dependent. This can be done by determining all points x D .x0 ; x1 ; x2 ; x3 / 2 S 3 for which a solution .a; b/ 2 R2 ; a2 C b 2 D 1, of      j x0 C x1 i C x2 j C x3 k D x0 C x1 i C x2 j C x3 k ai C bk exists. A necessary condition for solving this equation turns out to be x02 C x22 D x12 C x32 D

1 2

and the solution .a; b/ is given by a D 2.x1 x2 C x0 x3 /; b D 2.x2 x3  x0 x1 /: p p p p If we put 2x0 D cos , 2x2 D sin , 2x1 D cos and 2x3 D sin , then the numbers a and b can be expressed in the following form: a D sin. C /; b D  cos. C /: ˚  Let p.i; k/ D x 2 S 3 j x02 C x22 D 1=2 D x12 C x32 . It can be checked that p.i; k/ is invariant under the action of the group U.1/ to the left (and also to the right). Proposition 11.1.16. The operator GS degenerates on the set L .p.i; k// Š S 1 ; i.e., the vector fields dL .Xi / and dL .Xk / are linearly dependent (but do not vanish simultaneously). In order to make a comparison with the Heisenberg–Grushin operator GH , we shall present the spherical Grushin operator GS explicitly in two ways. First, we consider the map L given by L W H Š R4 ! R3 ; .x0 ; x1 ; x2 ; x3 / 7! .u1 ; u2 ; u3 /; u1 D x0 x1  x2 x3 ; u2 D x0 x3 C x1 x2 ; u3 D x02 C x22  1=2; 2 which does realize P 1 C as a sphere S.1=2/ of radius 1=2 in R3 . This map is characterized by the equation

   u21 C u22 C u23  1=4 D x02 C x22 x02 C x12 C x22 C x32  1 :

(11.1.10)

11 Heat Kernel for the Sub-Laplacian on the Sphere S 3

284

Using (11.1.10) it can be easily checked that L .p.i; k// coincides with the equator. Moreover, S 3 is divided into two connected components having the twodimensional torus p.i; k/ as the common boundary. As stated in the beginning of this section, we can descend the right actions of fe t i g and fe t k g onto R3 through the map L . The resulting actions are the rotations with respect to the u2 -axis and u1 -axis, respectively. The vector fields dL .Xi / and dL .Xk / are of the form @ @ C 2u3 ; @u3 @u1 @ @ dL .Xk / D 2u2 C 2u3 : @u3 @u2

dL .Xi / D 2u1

(11.1.11) (11.1.12)

It is obvious that dL .Xi / and dL .Xk / are tangent to the sphere and are linearly dependent only along the equator f .u1 ; u2 ; 0/ j u21 C u22 D 1=4 g. The spherical Grushin operator GS is the restriction of  1˚ 1 dL .Xi/ 2 C dL .Xk / 2  GS D 4 4 2

@2 @2 @ @ @ C  u1  2u3 D .u21 C u22 / 2 C u23 2 2 @u3 @u1 @u3 @u1 @u2  u2

@ @2 @2  2u1 u3  2u2 u3 @u2 @u1 @u3 @u2 @u3

to the sphere u21 C u22 C u23 D 1=4. Next, we shall provide an expression in local coordinates, with the exception of one point. Consider the following local trivialization: (11.1.13) ˆL W U.1/  C ! U0 ; p L W .; z/ D .; x C 1y/ 7! ! !  z Re p 1 C Re p i (11.1.14) 1 C jzj2 1 C jzj2 ! ! z  j  Im p k 2 S 3  H: C Im p 1 C jzj2 1 C jzj2 p Then the variable z D x C 1y is a local coordinate on the subset L .U0 /, and with this coordinate the vector fields dL .Xi / and dL .Xk / are expressed as  @  @ dL .Xi / D 1 C x 2  y 2 C 2xy ; @x @y  2  @ @ C x  y2  1 : dL .Xk / D 2xy @x @y

11.1 Sub-Riemannian Structure and Sub-Laplacian on the Sphere S 3

285

Only on the unit circle x 2 C y 2 D 1 are these vectors linearly dependent, and the spherical Grushin operator has the form  @2  GS D 1 C 2.x 2  y 2 / C .x 2 C y 2 /2 @x 2  @2  @ @ @2 C 4x C 4y : C 1  2.x 2  y 2 / C .x 2 C y 2 /2 C 8xy @y 2 @x@y @x @y Remark 11.1.17. Recently, in [3] a structure on a manifold which is called almostRiemannian was defined. Roughly speaking, this structure is given through a set of Lie bracket generating vector fields without assuming the linear independence at every point (the number of the vector fields will be less than the dimension of the manifold). Moreover, a nondegeneracy condition for the sub-manifold on which 2 the vector ˚ fields are linearly dependent is assumed. The S -case carrying the vector fields dL .Xi /; dL .Xk / is mentioned as an example. Now using the heat kernel expression for the sub-Laplacian on S 3 given in the last section, we give an integral expression of the heat kernel for the spherical Grushin operator. Since Ksub . tI x; y / satisfies the following invariance:   Ksub tI g  x; g  y ;

8g; x; y 2 S 3 ;

according to the construction of the spherical Grushin operator GS , the heat kernel   KGS . t; L .x/; L .y/ / 2 C 1 RC  P 1 C  P 1 C of GS can be expressed in the following integral form. Theorem 11.1.18.   1 KGS t; L .x/; L .y/ D p 1

Z

  d Ksub tI   x; y :  U.1/

We also have Theorem 11.1.19. The following trace formula holds:   tr e t GS Z Z  d  1 Ksub tI x 1    x; 1 dS.x/ D p 3  1 S U.1/  m 2 qC3 1   1 2 2 2 X X C2mC` D p e t 4m C4m.1C`/C2` 1 `D0 mD0 a2mC` Z Z h Z   imC`   jw0 .x/j2 C jw1 .x/j2 u0 C w0 .x/.  /w1 .x/ u1  U.1/ S 3 XS

11 Heat Kernel for the Sub-Laplacian on the Sphere S 3

286



h

   im mC` m  u 0  v0 jw0 .x/j2 C jw1 .x/j2 v0 C w0 .x/.  /w1 .x/ v1 p 2„j.u;v/j

 j.u; v/jq  e 

d S .u; v/ dS.x/  2 

 m qC3 1 1  2 2 2 X X C2mC` e t 4m C4m.1C`/C2` Cp 1 `D1 mD0 a2mC` Z Z Z h   im    jw0 .x/j2 C jw1 .x/j2 u0 C w0 .x/.  /w1 .x/ u1 U.1/



h

S3

XS

   imC` m mC`  u 0  v0 jw0 .x/j2 C jw1 .x/j2 v0 C w0 .x/.  /w1 .x/ v1 p 2„j.u;v/j

 j.u; v/jq  e 

S .u; v/ dS.x/

d ; 

where we have used the identities w0 . x 1    x / D jw0 .x/j2 C jw1 .x/j2 ;   w1 . x 1    x / D w0 .x/    w1 .x/: Based on the previous information, the zeta-regularized determinant of the subLaplacian and other related operators is computed; see [10]. Similar computations can be found in [10] for the case of a two-step, codimension-1 sub-Riemannian structure on the seven-dimensional sphere S 7 . The problem of finding the heat kernel in the case of the two-step, codimension-3 sub-Riemannian structure on S 7 still remains an open problem. In the case of the sphere S 7 , expressing the zeta-regularized determinant is not easy. However, we know that the spectral zeta function is holomorphic at the origin.

Chapter 12

Finding Heat Kernels Using the Laguerre Calculus

12.1 Introduction In this chapter, we are going to use a harmonic analysis method to construct the heat kernels and fundamental solutions of the sub-Laplacian on the Heisenberg group. This method relies on Laguerre calculus. We shall start with a beautiful idea of Mikhlin from his 1936 study of convolution operators on R2 . Let K be a principal value (P.V.) convolution operator on R2 : Z K.f /.x/ D lim

"!0 jyj>"

K.y/f .x  y/dy;

where f 2 C01 .R2 / and K 2 C 1 .R2 n f.0; 0/g/ is homogeneous of degree 2 with vanishing mean value; i.e., Z K.y/dy D 0: jyjD1

Thus we can write K.x/ D

f ./ ; r2

where f ./ D

x D x1 C ix2 D re i ; X

fm e i m :

m2Z;m¤0

Suppose that g is another smooth function on Œ0; 2 with g./ D

X

gm e i m :

m2Z;m¤0

Then g induces a principal value convolution operator G on R2 with kernel In [91], we found the following identity.

O. Calin et al., Heat Kernels for Elliptic and Sub-elliptic Operators, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-4995-1 12, c Springer Science+Business Media, LLC 2011 

g. / . r2

289

290

12 Finding Heat Kernels Using the Laguerre Calculus

Proposition 12.1.1. jmji jmj e i m jkji jkj e i k jm C ki jmCkj e i.mCk/  D : 2 2 r2 R 2 r2 2 r2 Here R2 stands for the standard convolution on R2 . X

Definition 12.1.2. Let f ./ D

fm e i m ;

m2Z;m¤0

inducing the principal value convolution operator K, with kernel Then the symbol .K/ of K is defined by .K/ D

X m2Z;m¤0

jmji jmj 2

f . / r2

on C01 .R2 /.

!1 fm e i m :

With this definition we may rewrite Proposition 12.1.1 as follows. Theorem 12.1.3. Let K and G be two principal value convolution operators on C01 .R2 /. Then      K R2 G D .K  .G : Now we shall generalize the result of Theorem 12.1.3 to the Heisenberg group. The nonisotropic Heisenberg group Hn is the Lie group with the underlying manifold Cn  R D fŒz; t W z 2 Cn ; t 2 Rg and the multiplication law " Œz; t  Œw; s D z C w; t C s C 2Im

n X

# aj zj w Nj ;

(12.1.1)

j D1

where a1 ; a2 ; : : : ; an are positive numbers. It is easy to check that the multiplication (12.1.1) does indeed make Cn  R into a group whose identity is the origin e D Œ0; 0, and where the inverse is given by Œz; t1 D Œz; t. The Lie algebra of Hn is a vector space which, together with a Lie bracket operation defined on it, represents the infinitesimal action of Hn . Let hn denote the vector space of left invariant vector fields on Hn . Note that this linear space is closed with respect to the bracket operation ŒV1 ; V2  D V1 V2  V2 V1 : The space hn , equipped with this bracket, is referred to as the Lie algebra of Hn .

12.2 Laguerre Calculus

291

The Lie algebra structure of hn is most readily understood by describing it in terms of the following basis: Xj D

@ @ C 2aj yj ; @xj @t

Yj D

@ @  2aj xj @yj @t

and T D

@ ; @t

(12.1.2)

where j D 1; 2; : : : ; n, z D .z1 ; z2 ; : : : ; zn / 2 Cn with zj D xj C iyj ; t 2 R. Note that we have the commutation relations ŒYj ; Xk  D 4aj ıjk T

for j; k D 1; 2; : : : ; n:

(12.1.3)

Next, we define the complex vector fields @ 1 @ i aj zj ZN j D .Xj Ci Yj / D 2 @Nzj @t

and Zj D

@ 1 @ Ciaj zNj .Xj i Yj / D 2 @zj @t (12.1.4)

for j D 1; 2; : : : ; n. Here, as usual, @ 1 D @zj 2



@ @ i @xj @yj

 and

@ 1 D @Nzj 2



@ @ Ci @xj @yj

 :

The commutation relations (12.1.3) then become ŒZN j ; Zk  D 2i aj ıjk T with all other commutators among the Zj , ZN k and T vanishing.

12.2 Laguerre Calculus Laguerre calculus is the symbolic tensor calculus on the Heisenberg group Hn . It was first introduced on H1 by Greiner [57] and extended to Hn and Hn  Rd by Beals, Gaveau, Greiner and Vauthier [17]. The Laguerre functions have been used in the study of the twisted convolution, or equivalently, the Heisenberg convolution for several decades (see Geller [51] and Peetre [96]). The Laguerre functions also played an important role in the Fock–Bargmann and Schr¨odinger representations of the Heisenberg group (see Folland [43] for details). But it was in Greiner [57] that for the first time Laguerre functions were connected with left invariant convolution operators on H1 , and were used to invert some basic differential operators on H1 , namely, the Lewy operator and the Heisenberg sub-Laplacian. In this chapter we shall use the Laguerre calculus to find the heat kernel and fundamental solution of the Kohn Laplacian and Paneitz operator. Of course, this method has some limitations since it depends heavily on the group structure and

292

12 Finding Heat Kernels Using the Laguerre Calculus

orthogonality of the Laguerre functions. However, this method allows us to construct heat kernels for higher-order partial differential operators, like powers of the subLaplacian and Paneitz operator on the group. In order to introduce the Laguerre calculus, we first recall the definitions of the twisted and P.V. convolutions on the Heisenberg group. Consider the functions f; g 2 C01 .Hn /. The Heisenberg convolution is given by Z

f .y/g.y1 x/dV .y/I

f  g.x/ D

(12.2.5)

Hn

here dV .y/ is the Haar measure on Hn and is exactly the Euclidean measure on R2nC1 .

12.2.1 Twisted Convolution We focus our attention on the phase space Rn  Rn , which we identify with Cn via  2 Cn ,  D u C i v $ .u; v/ 2 Rn  Rn . Let 2 3 a1 6 a2 7 6 7 AD6 7 :: 4 5 : an be a positive definite diagonal n  n real matrix. Consider the symplectic form < ;  > given by the Heisenberg group multiplication law (12.1.1) and defined by 0 N D 2Im @ < z; w > D 2Im.Az  w/

n X

1 aj zj wN j A ;

j D1

where z, w 2 Cn . With  a fixed real constant, we can define the twisted convolution of two functions F and G by Z .F  G/.z/ D

Cn

F .z  w/G.w/e i d wI

(12.2.6)

here d w is the Euclidean measure on Cn . Notice that, in view of the antisymmetry of < ;  >, we have < z  w; w > D  < w; z >; thus G  F D F  G; so the twisted product is not commutative.

12.2 Laguerre Calculus

293

The twisted convolution arises when we analyze the convolution of functions on the Heisenberg group in terms of the Fourier transform in the t-variable. To see this, let f .z; t/ be a test function on Hn . Define e  .z/ D f

Z

f .z; t/e it dt:

(12.2.7)

R

Similarly define e g  when g is another test function on Hn . Suppose f  g is the convolution of f and g on Hn . Then

B

e  e .f  g/ D f g :

(12.2.8)

12.2.2 P.V. Convolution Operators In order to show the regularity of the solution operator S.f / D f  ‰˛ , we need to introduce principal value convolution operators on Hn . These operators are the analog of Calder´on–Zygmund principal value convolution operators on Rn . As we know, the underlying manifold of Hn is R2nC1 ; but the role of the additive structure in R2nC1 is supplied by the Heisenberg group multiplication law (12.2.4). Moreover, the group law forces us to use nonisotropic dilations on Hn ; i.e., x 7! ı ı x D ı ı Œz; t D Œız; ı 2 t for all ı > 0. These dilations are automorphisms of the group Hn : ı ı .x  y/ D .ı ı x/  .ı ı y/I but the standard isotropic dilations of R2nC1 are not automorphisms of Hn . A function f defined on Hn is said to be H-homogeneous of degree m on Hn if f .ı ı Œz; t/ D f .ız; ı 2 t/ D ı m f .z; t/ for all ı > 0. For example, the fundamental solution ‰m of Lm ˛ is H-homogeneous of degree 2n C 2m  2. Next we introduce the norm function  given by 1=4  .x/ D kzk4 C t 2 ;

where kzk2 D

n X

aj jzj j2 :

(12.2.9)

j D1

Obviously, we have .x1 / D .x/ D .x/ and .ı ı x/ D ı.x/. In addition, the function  satisfies the triangle inequality: .x  y/  C1 f.x/ C .y/g

294

12 Finding Heat Kernels Using the Laguerre Calculus

for some universal constant C1 . The distance function, d.x; y/ of points x; y 2 Hn , is defined to be d.x; y/ D .y1  x/: It is clear that d.x; y/ satisfies the symmetric property d.x; y/ D d.y; x/. Suppose that K 2 C 1 .Hn n f0g/ is H-homogeneous of degree . Then K is locally integrable near the origin if > 2n  2. See Folland and Stein [45] for the proof of this statement. Definition 12.2.1. Let K 2 C 1 .Hn n f0g/ be H-homogeneous of degree 2n  2. Then K is said to have mean value zero property if Z K.x/d.x/ D 0;

(12.2.10)

.x/D1

where d.x/ is the induced measure on the Heisenberg unit sphere .x/ D 1. Using Theorem 3 and Corollary 5.24 of Chap. XII in Stein [105], we can get the basic estimate concerning P.V convolution operators on Hn : Theorem 12.2.2. Let K 2 C 1 .Hn n f0g/, H-homogeneous of degree 2n  2 with mean value zero property. Then K induces a principal value convolution operator, given by Z K.f /.x/ D .f  K/.x/ D lim

"!0 d.x;y/>"

f .y/K.y1  x/dV .y/;

(12.2.11)

for f 2 C01 .Hn /. Moreover, the operator K given by (12.2.11) can be extended to a bounded operator from the Lp -Sobolev space Lpk .Hn / into itself, for 1 < p < 1 and k 2 ZC .

12.2.3 Laguerre Functions .˛/

Consider the generalized Laguerre polynomials Lk .x/ defined by their usual generating function formula 1 X kD1

.˛/

Lk .x/wk D

n xw o 1 exp  ; for ˛ D 0; 1; 2; : : : ; x  0; .1  w/˛C1 1w and jwj < 1:

(12.2.12)

12.2 Laguerre Calculus

295

Definition 12.2.3. Let z D jzje i and k; p D 0; 1; 2; : : :. Then we define  1=2

.k C 1/ 2 .p/ e .p/ .z; / D 2jj W .2jjjzj2 /p=2 e ip e jjjzj Lk .2jjjzj2 /; k  .k C p C 1/ (12.2.13) 1=2 

.k C 1/ e .p/ .z; / D 2jj .1/p W k 

.k C p C 1/ 2

 .2jjjzj2 /p=2 e ip e jjjzj L.p/ .2jjjzj2 /: k

(12.2.14)

12.2.4 Laguerre Calculus on H1 e .p/ The most important property of W k .z; / is the following theorem of Greiner [57]: Theorem 12.2.4. Let p; k; q; m D 1; 2; : : :. Then e .pk/ jj W e .qm/ D ı .q/  W e .pm/ ; W .p^k/1 .q^m/1 k .p^m/1 .p/ e .qk/ e .qm/ D ım e .pk/ jj W W  W .q^k/1 ; .p^k/1 .q^m/1

(12.2.15)

.q/

where a ^ b D min.a; b/ and ık denotes the Kronecker delta function; i.e., ık.q/ D 1 if k D q and vanishes otherwise. e .p/ is another function of the Thus the twisted convolution of two functions of W k same type. This surprising result justifies the use of Laguerre function expansion on the Heisenberg group in analogy with Mikhlin’s use of the spherical harmonics on Rn . .p/ Let Wk .z; t/; ˙ p; k D 0; 1; 2; : : :, be the inverse Fourier transform of .p/ e .z; / with respect to  i.e., W k Wk.p/ .z; t/ D

1 2

Z

1 1

e .p/ .z; /d : ei t  W k

These are the kernels of the generalized Cauchy–Szeg¨o operators on H1 . In particular, .0/

.0/

.0/

W0 .z; t/ D WC;0 .z; t/CW;0 .z; t/ D SC CS ; where S˙ D

1 1   2 .jzj2 i t/2 (12.2.16)

denotes the Cauchy–Szeg¨o kernels in H1 . The following result implies that the generalized Cauchy–Szeg¨o kernels indeed induce principal value convolution operators.

296

12 Finding Heat Kernels Using the Laguerre Calculus .p/

Theorem 12.2.5. The generalized Cauchy–Szeg¨o kernels Wk .z; t/ are in C 1 .H1 n f0g/ and have zero mean value property. Formula (12.2.15) is reminiscent of matrix multiplication. To show this similarity, Greiner [57] introduced the Laguerre matrix: Definition 12.2.6. We define the positive Laguerre matrix  e .pk/ MC W .p^k/1 e of W to be the infinite matrix which has one at the intersection of the .p^k/1 pth row and kth  column and zeros everywhere else. The negative Laguerre matrix .pk/ e M W .p^k/1 can be defined as the transpose of the positive Laguerre matrix. .pk/

Following this definition, Theorem 12.2.4 takes the following form:   .pk/  .qm/ .qm/ e .q^m/1 e .pk/ jj W e .p^k/1  MC W e .q^m/1 : MC W D M W C .p^k/1 (12.2.17) The Laguerre matrix for any left invariant convolution operator can be defined as in [57] and [17]. In particular, we can define the Laguerre matrix for any left invariant differential operator on H1 , since we can write it in the form of convolution operators. We omit the details here, and the interested reader can consult [17] and Berenstein–Chang–Tie [19]. We only list the following results: Q Theorem 12.2.7. If F and G are two P.V. convolution operators on H1 , and M.F/ Q and M.G/ denote the Laguerre matrices of F and G, respectively, then e D M.e e D MC .e e ˚ M .e e (12.2.18) M.e F  G/ F/  M.G/ F/  MC .G/ F/  M .G/: A simple consequence of this theorem is Corollary 12.2.8. Let I denote the identity operator on C01 .H1 /. Then I is induced .p/ by the identity Laguerre matrix W˙ .e I/ D .ık /.

12.2.5 Laguerre Calculus on Hn We define the n-dimensional version of the exponential Laguerre functions on Hn by the n-fold product: e .p/ .z; / D W k

n Y j D1

e .pj / .paj zj ; /; aj W kj

e .pj / .paj zj ; / is given by (12.2.13). where W kj

(12.2.19)

12.2 Laguerre Calculus

297

Composing two functions of the type (12.2.19) via twisted convolution yields 2 e .p/  W e .q/ 4 W m D k

n Y

3 .pj /

e aj W k

j

j D1

D

n Y j D1

aj2

Z

2

p . aj zj ; /5  4

3 j/ p e .q 5 aj W mj . aj zj ; /

j D1 .pj /

R2

n Y

e e 2i aj Im.zj wN j / W k

j

p j/ e .q . aj .zj  wj /; /W mj

p . aj wj ; /d wj n Y j/ p e .pj /  W e .q D aj W mj . aj zj ; /: k j D1

j

Consequently, Theorem 12.2.4 implies that Theorem 12.2.9. Let kj ; pj ; mj and qj D 1; 2; 3; : : : for j D 1; 2; : : : ; n. Then e .qm/ D ı .q/  W e .pm/ ; e .pk/ jj W W k .k^p/1 .m^q/1 .p^m/1 .p/ e .qk/ e .pk/ jj W e .qm/ D ım  W .q^k/1 ; W .k^p/1 .m^q/1

(12.2.20)

where k D .k1 ; k2 ; : : : ; kn /; p D .p1 ; p2 ; : : : ; pn /; m D .m1 ; m2 ; : : : ; mn / and q D .q1 ; q2 ; : : : ; qn /. Here .k ^ p/  1 D .min.p1 ; k1 /  1; : : : ; min.pn ; kn /  1/ .p/

and ık D

Qn

.pj / j D1 ıkj

is the n-fold Kronecker delta function.

Instead of the Laguerre matrix, Beals et al. [17] introduced the definition of a Laguerre tensor for the convolution operators on Hn . We omit the details here. Then we have the n-fold version of Theorem 12.2.7: Theorem 12.2.10. (The Laguerre calculus on Hn ) Let F and G induce the convolue denote the Laguerre tensors of F and tion operators on Hn . Let M.e F/ and M.G/ e e e e G, respectively. Then M.F  G/ D M.F/  M.G/. Corollary 12.2.11. The identity operator I on C01 .Hn / is induced by the identity Laguerre tensor: I/ D .ık.p1 1 /    ık.pnn / /: M˙ .e

12.2.6 Left Invariant Differential Operators A left invariant differential operator P on Hn is a polynomial P.X; Y; T/ with constant coefficients, or in complex coordinates, a polynomial in vector fields T, Zj ,

298

12 Finding Heat Kernels Using the Laguerre Calculus

and ZN j . We can have the following representation for P as a convolution operator on Hn : P D PI D

1 X jkjD0

PWk.0;:::;0/ ; ;:::;kn 1

where I D

1 X jkjD0

Wk.0;:::;0/  ;:::;kn 1

(12.2.21)

is the identity operator on C01 .Hn /. In particular, T, Zj , and ZN j , j D 1; 2; : : : ; n; can be represented as convolution operators and written in the Laguerre tensor forms. This is contained in the next proposition: Proposition 12.2.12. (1) M.e T/ is the i multiple of the identity Laguerre tensor: .p / .p / M˙ .e T/ D i.ık1 1 : : : ıknn /:

(2) Zj ; j D 1; 2; : : : ; n, has the following Laguerre tensor representation: M.e Zj / D MC .e Zj / ˚ M .e Zj /; where q .p / .p C1/ ;:::;pn / M .e Zj /k.p11;:::;k D 2aj pj jjık1 j : : : ık j : : : ık.pnn / n j

and MC .e Zj / D M .e Zj /t : N j , j D 1; 2; : : : ; n, has the following Laguerre tensor representation: (3) Z N j / D M.e Zj /t : M.e Z N T/ D P.Z1 ; : : : ; Zn ; ZN 1 ; : : : ; ZN n ; T/ denote a Theorem 12.2.13. Let P D P.Z; Z; left invariant differential operator on Hn i.e., P is a polynomial in the vector fields T, Zj , and ZN j , j D 1; 2; : : : ; n. Then e D P.M.e N i /; M.P/ Z/; M.e Z/;

(12.2.22)

N D .M.e Zn // and M.e Z/ ZN 1 /; : : : ; where we set M.e Z/ D .M.e Z1 /; : : : ; M.e e N n //. M.Z Example 12.2.14. Assume that n D 1 and a1 D 1. Then we have 2

0 6 p 60 MC .e Z1 / D 2jj 60 4 :: : and

p

1 p0 0 2 p0 0 0 0 3 :: :: :: : : :

t Z1 / D MC .e Z1 / : M .e

3    7 7 ;   7 5 :: :

12.2 Laguerre Calculus

299

Now we may set 2

0

6 p1 6 1 1 6 6 0 MC .K/ D p 6 2jj 6 6 0 4 :: : and

0 0 p1 2

0 :: :

3 0  0   7 7 7 0   7 7; 7 0   7 5 :: : : : :

0 0 0 p1 3

:: :

t M .K/ D MC .K/ :

Thus 1 X 1 e .1/ e ˙ .z; / D p 1 p K W ˙;k .z; /: 2jj kD0 k C 1 .1/

e ˙;k .z; /, we sum the series Using the definition of W 2

jjjzj e / D 2jjze K.z; 

Z

1 1X 0

.1/

r k Lk .2jjjzj2 /dr:

kD0

Since 1 X

r k L.1/ .x/ D k

kD0

x ex e  1r ; .1  r/2

therefore, 2

jjjzj e / D 1  e K.z; ;  zN

and 1 K.z; t/ D 2 2 zN

Z

C1

1

2

e i t jjjzj d  D

z :  2 .jzj4 C t 2 /

Hence, we recover the Greiner, Kohn and Stein theorem [60]: .0/ D I  S ; Z1 K D I  W;0 .0/ KZ1 D I  WC;0 D I  SC :

Here S˙ are the Cauchy–Szeg¨o kernels which were defined by (12.2.16).

300

12 Finding Heat Kernels Using the Laguerre Calculus

12.3 The Heisenberg Sub-Laplacian The Heisenberg sub-Laplacian is the differential operator L˛ D 

n 1X .Zj ZN j C ZN j Zj / C i ˛T 2

(12.3.23)

j D1

with Zj and ZN j given by (12.1.4). In the case of aj D 1 for all j , the operator L˛ was first introduced by Folland and Stein [45] in the study of @N b complex on a nondegenerate CR manifold. They found the fundamental solution of L˛ . Beals and Greiner [18] solved the case when the aj are different. Next we consider two operators related to L˛ . The first one is the powers of L˛ . In the special case of aj D 1=4 for all j D 1; 2; : : : ; n and ˛ D 0, Lm 0 , 1  m  n, has been studied by Dolley, Benson and Ratcliff [40]. If all the aj are equal, the unitary group U.n/ acts on Hn via uŒz; t D Œu.z/; t

for u 2 U.n/;

Œz; t 2 Hn :

(12.3.24)

L˛ is invariant under the U.n/-action. The key idea in [40] is to exploit this invariance by expanding the fundamental solution in terms of U.n/-spherical functions ;m which are given by 2 =4

;m .z; t/ D e i t e jjjzj

.n1/

Lm1 .j jjzj2 =2/

.n1/

with 2 R n f0g, m 2 ZC . Here Lm denotes the generalized Laguerre polynomial. However, the invariance (12.3.24) does not hold for the nonisotropic case. Instead, we shall apply the Laguerre calculus to solve the general case. Since the Laguerre tensor of Lm ˛ is simply diagonal, its inverse is also simple and diagonal. As usual, taking the partial Fourier transform with respect to the t-variable, one has 3 2 1 n n X X Y 1 e e .0/ .paj zj ; / : 4 ZN j C e L˛ D e L˛e .e Zj e ZN j e aj W ID Zj /  ˛ 5 kj 2 jkjD0

j D1

j D1

(12.3.25) A simple calculation yields 1 e .0/ .paj zj ; /: (12.3.26) e .0/ .paj zj ; / D .2k C 1/jjaj W ZN j C e ZN j e  .e Zj e Zj /W k k 2 Thus (12.3.25) and (12.3.26) imply 1 0 1 n n X X Y e e .0/ .paj zj ; / : @ L˛ D .2kj C 1/jjaj  ˛ A aj W kj jkjD0

j D1

j D1

(12.3.27)

12.3 The Heisenberg Sub-Laplacian

301

Consequently, the Laguerre tensor of the convolution operator induced by L˛ is 02

n X

M.e L˛ / D jj @4

3

1

.p / .p / .2kj C 1/aj  ˛sgn./5 ık1 1    ıknn A ;

(12.3.28)

j D1

which is invertible as long as ˛ does not belong to the exceptional set ƒ˛ , where ƒ˛ D

n X

.2kj C 1/aj ;

k D .k1 ; k2 ; : : : ; kn / 2 .ZC /n :

j D1

According to Theorem 12.2.10, the inverse Laguerre tensor of (12.3.28) is 02 1 31 n X 1 B C .2kj C 1/aj  ˛sgn./5 ık.p1 1 /    ık.pnn / A : (12.3.29) M.e L˛ / D jj1 @4 j D1

If we write it in the Laguerre series expansion: 11 0 n n 1 X X Y 1 e e .0/ .paj zj ; /; A @ .z; / D jj .2k C 1/a  ˛sgn./ aj W L1 j j ˛ k j D1

jkjD0

j D1

j

(12.3.30) then we can sum this series in the sense of Abel, take the inverse Fourier transform with respect to , and find the fundamental solution of L˛ . In fact, the above calculation makes sense for any polynomial of L˛ . We will carry out this extension, and N find the fundamental solution of Lm ˛ and the Paneitz operator L˛ L˛ , in the rest of this chapter. Remark 12.3.1. Being concerned about any possible confusions regarding the formulas (12.3.25) and (12.3.27), we shall clarify the notation  one more time. The operators L˛ and P˛ are left invariant. So when they act on a function f , we may write L˛ .f / and P˛ .f /. Taking the partial Fourier transform with respect to f˛  fQ; and P f˛  fQ. Here the t-variable (whose dual variable is ), one has L  means “twisted convolution,” concept defined in this chapter. So, removing the f˛  and P f˛  . We may decompose function variable fQ, the operators become L these operators into Laguerre tensor expansions, which are given by (12.3.25) and (12.3.27). Before going further, we shall say a few words about the exceptional set ƒ˛ of the operator L˛ . Basically, we look at the exceptional set from a different point of view. From the calculations of this section we know that ˛ ¤ ˙

n X

.2kj C 1/aj W

j D1

k D .k1 ; k2 ; : : : ; kn / 2 ZnC :

302

12 Finding Heat Kernels Using the Laguerre Calculus

Let us consider the “real part” of the operator L˛ : L D 4L0 D 

2n X

Xj2

j D1

of the Kohn Laplacian and the operator T D @t@ . Since L and iT are essentially self-adjoint strongly commuting operators, there is a well-defined joint spectrum. We define the joint spectrum of the pair .L; iT / as the complement of the set of . ; / 2 C2 for which there exists Lp -bounded operators A and B with A. I  L/ C B. I  iT / D I: This implies that the spectrum should be the set of . ; / 2 C2 for which neither I  L nor I  iT is invertible. From the Laguerre calculus, we know that a convolution operator is invertible if and only if its Laguerre tensor is invertible. So we reduce the invertibility of an operator to that of its Laguerre tensor. In fact, we know that the Laguerre tensor of the operator I  L is 0 1 n X

 .p1 / M. e Ie L/ D @  jj aj .2k1 C 1/ ık1    ık.pnn / A j D1

and the Laguerre tensor of the operator I  iT is 

e/ D C ı .p1 /    ı .pn / : M. e I  iT k1 kn Therefore, . I  L/

is invertible ,  jj

n X

aj .2k1 C 1/ ¤ 0

j D1

for all k 2 .ZC /n ,  2 R and . I  iT / is invertible , C  ¤ 0: Hence the spectrum of L alone is the set of nonnegative numbers f 2 R W  0g, the spectrum of iT is the set of real numbers R, and the joint spectrum of .L; iT / is the union of 8 9 n < = X . ; / 2 C2 W D j j .2kj C 1/aj and 2 R : ; j D1

over the set k 2 .ZC /n . We can also write the joint spectrum in the form ( ) [ ˙ . ; / 2 C2 W  0; D Pn : .L; iT / D j D1 .2kj C 1/aj n k2.ZC /

(12.3.31) The set (12.3.31) (see Fig. 12.1) is called the Heisenberg brush (see [104]).

12.4 Powers of the Sub-Laplacian

303

Fig. 12.1 The Heisenberg brush in the isotropic case a1 D    D a n D a

12.4 Powers of the Sub-Laplacian In this section we consider the powers of the Heisenberg sub-Laplacian Lm ˛ . For 1  m  n, this work has been done by Dolley, Benson and Ratcliff [40] in the case when all the aj are equal. They used a different method. First we will find the Laguerre tensor of the operator Lm ˛ from the Laguerre tensor of L˛ . Similarly to (12.3.25), we can take the Fourier transform with respect to t, and write e Lm ˛ as a twisted convolution form: e eme Lm ˛ D L˛ I D

2

3m n n X Y 1 e .0/ .paj zj ; / : 4 Zj /  ˛ 5 .e Zj e ZN j e aj W ZN j C e kj 2

1 X

j D1

jkjD0

j D1

(12.4.32) Then (12.3.26) yields e Lm ˛ D

1 X jkjD0

0 @

n X

1m .2kj C 1/jjaj  ˛ A

j D1

n Y j D1

e .0/ .paj zj ; / : (12.4.33) aj W kj

Consequently, the Laguerre tensor of the convolution operator induced by Lm ˛ is 02 m M.e L˛ / D jjm @4

n X

3m

1

.p / .p / .2kj C 1/aj  ˛sgn./5 ık1 1    ıknn A ;

(12.4.34)

j D1

which is invertible as long as ˛ does not belong to the singular set ƒ˛ , where ƒ˛ D

8 < :

˙

n X

.2kj C 1/aj W

j D1

k D .k1 ; k2 ; : : : ; kn / 2 ZnC

9 = ;

:

304

12 Finding Heat Kernels Using the Laguerre Calculus

According to Theorem 12.2.10, the inverse Laguerre tensor of (12.6.54) is 02 m M.e L˛ / D jjm @4

n X

1

3m

.p / .p / ık1 1    ıknn A ;

.2kj C 1/aj  ˛sgn./5

j D1

(12.4.35) em .z; / in the following Laguerre series expansion: and we write its kernel ‰ 1m 0 n n 1 X X Y m e e .0/ .paj zj ; /: A @ .2kj C 1/aj  ˛sgn./ aj W ‰m .z; / D jj k j D1

jkjD0

j D1

j

(12.4.36) To find the fundamental solution of Lm ˛ , we can sum this series and take the inverse Fourier transform with respect to . First we introduce the following integral representation of Am : 1 1 D m A

.m/

Z

1

s m1 e As ds

for Re.A/ > 0:

(12.4.37)

0

Then, we can write (12.6.56) in the following form: 1 Z 1 m X Pn em .z; / D jj ‰ s m1 e . j D1 .2kj C1/aj ˛sgn.//s ds

.m/ 0 jkjD0



n Y

e .0/ .paj zj ; /: aj W k

(12.4.38)

j

j D1

Next, we interchange the summation and the integration, and use the definition e .0/ : of W kj

em .z; / ‰ jjm D

.m/

Z

1 0

jjnm D n  .m/ 

n Y j D1

s m1

1 X

Pn

e .

j D1 .2kj C1/aj ˛sgn./

/s ds

j D1

jkjD0

Z

1

s 0

m1

n Y

1 X

e .

Pn

jkjD0 2

2aj e aj jjjzj j L.0/ .2aj jjjzj j2 / k j

/s ds

j D1 .2kj C1/aj ˛sgn./

e .0/ .paj zj ; / aj W k j

12.4 Powers of the Sub-Laplacian

D

jjnm  n .m/ 

1 X

Z

1

305 n Y

s m1 e ˛sgn./s

0

2

2aj e aj saj jjjzj j

j D1

.e 2aj s /kj L.0/ .2aj jjjzj j2 /ds; k j

kj D0

Applying the generating formula for the Laguerre polynomials 1 X



1 xz D exp  .1  z/pC1 1z

L.p/ .x/zk k

kD0

em .z; /, we obtain to the last formula for ‰ Z

nm e m .z; / D jj ‰  n .m/

1

s

m1 ˛sgn./s

e

0



n Y 2aj e aj s 1  e 2aj s j D1

  2e 2aj s  exp aj jjjzj j 1 C ds 1  e 2aj s 3 2 n nm Z 1 Y jj aj 5 D n s m1 e ˛sgn./s 4  .m/ 0 sinh.aj s/ 2

j D1

8
0

and Re.m/ > 0

0

to give another derivation of the fundamental solution of Lm ˛ , 0  m  n. We can first write the inverse formally as Lm ˛ .z; t/ D

1

.m/

Z

1

s m1 e sL˛ ds

0

1 D nC1 

.m/

Z

1

s

mn2

0

Z

C1

./ n e 2˛

2 s

g.Iz;t /

dds:

1

By Fubini’s theorem, if z ¤ 0, we may interchange the order of integration. Moreover, if Re.m/ > 0, one has Z

1

˛

s mn e  s

0

ds D s2

Z

1

s nm e ˛s ds D

0

.n  m C 1/ .n  m/Š D nmC1 ; ˛ nmC1 ˛

so that we may rewrite Lm ˛ .z; t/ as 2m .n  m/Š ‰m .z; t/ D .2/nC1 .m/

Z

1

1

./ m1 e 2˛ d : Œg.I z; t/nmC1

(12.5.47)

This is exactly (12.4.43) obtained in the last section. When m > n, we cannot apply Fubini’s theorem since the integral in the variable s diverges as s ! 1. Similarly, we may consider the kernel induced by the operator ‰ˇ .z; t/ D Liˇ ˛ ı0 . We may combine the heat kernel in (12.5.46) with the following formula: wiˇ D

1

.iˇ/

Z

1

s iˇ 1 e sw ds;

Re.w/ > 0

and Im.ˇ/ > 0:

0

This leads to Liˇ ˛ .z; t/

.n C iˇ C 1/ D nCiˇ C1 2

.iˇ/

Z

1 1

.s/s iˇ 1 e 2˛ ds: Œg.sI z; t/nCiˇ C1

(12.5.48)

12.6 Fundamental Solution of the Paneitz Operator

309

By analytic continuation, formula (12.5.48) remains true in the case of real ˇ. This operator was studied by M¨uller and Stein [93] in the case of aj D 1 for all j D 1; : : : ; n and ˛ D 0. They have shown that the weak-type (1,1)-norm of the operator .2nC1/=2 as jˇj ! 1. Liˇ 0 is bounded from below by C jˇj We shall mention another application of formulas (12.5.47) and (12.4.43). 1=2 Replacing m in (12.5.47) by 12 , we obtain the kernel for L0 . Then kernels for the 1

1

  Riesz transforms Rj D Zj L0 2 and RnCj D ZN j L0 2 , j D 1; : : : ; n, which were first defined by Christ and Geller [36], can be obtained by composing Zj or ZN j to L12 0 . Here are the formulas:

1

Zj L0 2 D

.n C 12 /aj zNj 1

Z

3

2nC 2  nC 2

1

1

.s/s nC1 Œcoth.2aj s/ C 1 ds Œsg.sI z; t/nC3=2

(12.5.49)

.s/s nC1 Œcoth.2aj s/  1 ds: Œsg.sI z; t/nC3=2

(12.5.50)

and 1

 ZN j L0 2 D

.n C 12 /aj zj 2

nC 1 2



nC 3 2

Z

1 1

It is easy to see that these kernels are H-homogeneous of degree 2n  2 and satisfy the mean value zero property because the factor zN j or zj appears (see the next section for details).

12.6 Fundamental Solution of the Paneitz Operator In this section, we consider the Paneitz operator P˛ on the Heisenberg group, which is defined as 32 2 n X 1 P˛ D L˛ LN ˛ D 4 .Zj ZN j C ZN j Zj /5 C ˛ 2 T2 : 4

(12.6.51)

j D1

We shall about ˚ first recall ˚ n some background n the Paneitz operator. Consider the coframe ; !j ; !N j j D1 dual to T; Zj ; ZN j j D1 . Let ' 2 C01 .Hn / be a smooth function with compact support. It is easy to see that Ln ' D 

n n X 1X N j Zj / C .i nT/.'/ D  ZN j Zj .'/: .Zj ZN j C Z 2 j D1

j D1

310

12 Finding Heat Kernels Using the Laguerre Calculus

It follows that Pn ' D Ln LNn ' " n # i h 1 X D .Zj ZN j C ZN j Zj / C 2i nT .Zk ZN k C ZN k Zk /  2i nT ' 4 j;kD1

9 8 2 32 > ˆ n = mN j˛j : ˇ'j.ˇ / .tI x; /ˇ  j'1 jl0 Cj˛jCjˇ j B˛;ˇ .j  1/Š Proof of Estimation II: Note that '1 .tI x; / D rN .tI x; /, We can apply Theorem 15.1.4 for the multi-product ˆ1 .sj 1  sj /: For any l, there exists l0 such that  j 1 Z t Z s1 Z sj 2 / .mN / .0/ j  C j' j j  dsj 1    ds2 ds1 j' j'j .t/j.mN 1 l 1 l l 0

0

 j 1 / C j'1 j.0/  C j'1 j.mN l0 l0

0

0

0

t j 1 : .j  1/Š

mN by Proof of Theorem 15.2.1: By Estimation II, we can define '.t/ 2 S1;0 1 X

'j .t/ D '.t/:

j D0

Then '.tI x; D/ D ˆ.t/ satisfies

Z

t

RN .t/ C ˆ.t/ C

RN .t  s/ˆ.s/ds D 0; 0

and

ˇ ˇ ˇ .˛/ ˇ .mN / <  >mN j˛j expŒB˛;ˇ t: ˇ'.ˇ / .t/ˇ  C j'1 jl0

mNQ N For any N , choose NQ D N C m. Then for rNQ .t/ 2 S1;0 D S1;0 ; we can write

u.t/ D

Q 1 N X

uj .t/ C rNQ .t/

j D0

D

N 1 X j D0

uj .t/ C

Q 1 N X j DN

D gN .t/ C rN .t/;

uj .t/ C rNQ .t/

376

15 The Pseudo-Differential Operator Technique

P Q 1 N where rN .t/ D jNDN uj .t/ C rNQ .t/. It is clear that rN .t/ belongs to S1;0 , and hence we have arrived at the desired result.  Remark 15.2.2. The kernel of the operator U.t/ D u.tI x; D/ is given by the integral Z K.t; x; y/ D .2/n e i.xy/ u.tI x; /d: Rn

For instance, in the case of the Laplacian P D 2

and u.t/ D e jj t .

Pn

@2 j D1 @xj 2 ,

the symbol is p D jj2 ,

15.3 Basic Results for Pseudo-Differential Operators of Weyl Symbols We shall start with the definition of pseudo-differential operators of Weyl symbols, which will play an important role in the sequel. Definition 15.3.1. Let 0  ı    1; ı < 1. A pseudo-differential operator P on m Rn of Weyl symbol p.x; / 2 S;ı is defined by the formula P u.x/ D p w .x; D/u.x/ Z D Os  .2/n

 y  e iy p x C ;  u.x C y/dy d: 2 Rn Rn  Z xCy e i.xy/ p ;  u.y/dy d: D Os  .2/n 2 Rn Rn

In the rest of this chapter we shall use pseudo-differential operators of Weyl symbols. The reader can find details in the references H¨ormander [67]. and Iwasaki and Iwasaki [72].

15.3.1 Calculus with Pseudo-Differential Operators In this section we shall deal with the product formula, multi-product formula and estimates for symbols. The product of two pseudo-differential operators of Weyl

15.3 Basic Results for Pseudo-Differential Operators of Weyl Symbols

377

symbols P D p w .x; D/ and Q D q w .x; D/ is also a pseudo-differential operator of Weyl symbols given by  w .p w .x; D/q w .x; D// D p ıw q; that is, p w .x; D/q w .x; D/ D .p ıw q/w .x; D/: Expressed as an oscillatory integral, p ıw q is given by .p ıw q/.x; / D Os  .2/2n

Z

Z Rn Rn

Rn Rn

e i.y

1  1 Cy 2  2 /

  y2 y1 p x ;  C 1 q x C ;  C 2 dy 1 d1 dy 2 d2 : 2 2 In fact, p ıw q can be given in the form provided by the following theorem. m1 m2 Theorem 15.3.2. Let p 2 S;ı and q 2 S;ı . Then for any integer N , we have the expansion N 1  X 1 j w j .p; q/ C rN .p; q/; p ıw q D 2i j D0

where j .p; q/ D

X j˛jCjˇ jDj

.1/jˇ j .˛/ .ˇ / m.ı/j .x; / 2 S;ı ; p .x; /q.˛/ ˛ŠˇŠ .ˇ /

m.ı/N w rN .p; q/ 2 S;ı and there exist `0 and C such that the following estimate holds for any `: .m.ı/N /

w j` jrN

C

X

.˛/ .m j˛jCıjˇ j/

j˛jCjˇ jDN

jp.ˇ / j`C`1 0

.ˇ / .m Cıj˛jjˇ j/

jq.˛/ j`C`2 0

:

Remark 15.3.3. Pseudo-differential operators of Weyl symbols are pseudodifferential operators. In fact, we have p w .x; D/ D q.x; D/ if q.x; / D Os  .2/n

Z

and p.x; / D Os  .2/

n

  y e iy p x C ;  C  dy d 2 Rn Rn

Z

  y e iy q x C ;  C  dy d: 2 Rn Rn

m m is equivalent to q.x; / 2 S;ı . This shows that the condition p.x; / 2 S;ı

For the terms j .p; q/ given in Theorem 15.3.2, we have the following remark.

378

15 The Pseudo-Differential Operator Technique

Remark 15.3.4. It is clear that j .p; q/ D .1/j j .q; p/ for any integer j : So we have j .p; p/ D 0 if j is an odd integer: Remark 15.3.5. In particular, j .p; q/, j D 1; 2, have the following nice representation that is used in the construction of the fundamental solution: 1 .p; q/ D < J rp; rq >; 1 2 .p; q/ D  tr.JH p JH q /; 2 where rp Dt

 @ @ @ @  p; : : : ; p; p; : : : ; p ; @x1 @xn @1 @n

J is the 2n  2n matrix defined by 

0 I I 0

J D

;

and Hp denotes the Hessian matrix. That is,  Hp D

@x @x p @ @x p

@x @ p @ @ p

:

Proof of Theorem 15.3.2: In order to show the formula for p ıw q, we shall start by writing p w .x; D/q w .x; D/u.x/ Z 2n D Os  .2/ p

 x C x1 2

Z

Rn Rn

Rn Rn

e i.xx

1 / 1 Ci.x 1 x 2 / 2

  x 1 C x2  ; 1 q ;  2 u.x 2 / dx1 d 1 dx2 d 2 : 2

The integrand can be written as an oscillatory integral: p

 x C x1 2

Z  ;  1 D Os  .2/n

Rn Rn

e i.x

1 w/ 2

p

By the change of variables w D x2  y 2;

x1 D x C y 1;

x C w 2

 ;  1 dw d2 :

15.3 Basic Results for Pseudo-Differential Operators of Weyl Symbols

379

we have .x  x 1 /   1 C .x 1  x 2 /   2  .x 1  w/  2 D .x  x 2 /. 2  2 /  y 1 .2 C  1   2 /  y 2 2 D .x  x 2 /  y 1 1  y 2 2 if we set  D  2  2 ;

1 D 2 C  1   2 :

Then we obtain the following expression: p w .x; D/q w .x; D/u.x/ Z D Os  .2/3n

Z

Rn Rn

p

 x C x2  y 2 2

Z Rn Rn

Rn Rn

e i.xx

2 / i.y 1  1 Cy 2  2 /

  x C x2 C y 1  ;  C 1 q ;  C 2 u.x 2 / dy1 d1 dy2 d2 dx2 d: 2

So we have 2n

Z

Z

1

1

2

2

e i.y  Cy  / .p ıw q/.x; / D Os  .2/ Rn Rn Rn Rn   y2 y1 1 2 p x ; C  q x C ;  C  dy1 d1 dy2 d2 : 2 2 Using the Taylor expansion, we can write .p ıw q/.x; / D Os  .2/2n

Z Rn Rn

Z Rn Rn

e i.y

1  1 Cy 2  2 /

! ! y1 y2 1 2 ; C q x C ;  C  dy1 d1 dy2 d2 p x  2 2 Z Z 1 1 2 2 D Os  .2/2n e i.y  Cy  / n n n n R R R R ! ! X 1 y1 y2 .˛/  p ;  .1 /˛ q x C ;  C 2 dy1 d1 dy2 d2 x ˛Š 2 2 j˛j e.t/; 1 2 .p; e.t// D 2 .p; '/e.t/ C < J r'; Hp J r' > e.t/ 2 1 D ftr.JHp JH' /C < J r'; Hp J r' >ge.t/: 2 So we have 2  X 1 j j .p; e.t// 2i j D0   1 1 1 D p < J rp; r' >  tr.JH p JH ' /  < J r'; Hp J r' > e.t/: 2i 8 8

Neglecting the terms j .p; e.t// for j  3, we get the following equation for ' with '.0/ D 0: @ 1 1 1 'pC < J rp; r' > C tr.JH p JH ' / C < J r'; Hp J r' >D 0: @t 2i 8 8 It might not be easy to find a solution of the above equation. But neglecting the derivatives of p and ' of order of greater than 3, we can find a suitable solution as in the following. Using r.< J rp; r' >/ D Hp J r' C H' J rp; r.< J r'; Hp J r' >/ D r.< r'; JH p J r' >/ D 2H' JH p J r'; we have 1 1 @ r'  rp C fHp J r' C H' J rpg  H' JH p J r' D 0 @t 2i 4 and

@ 1 1 H'  Hp C fHp JH ' C H' JH p g  H' JH p JH ' D 0: @t 2i 4 The above equation means that X D iJH ' satisfies @ 1 1 X  A C .AX  XA/ C XAX D 0; @t 2 4

X jt D0 D 0;

15.4 Fundamental Solution by Symbolic Calculus: The Degenerate Case

385

with A D iJH p . The unique solution of the equation is X D 2 tanh.At=2/: So we can write @ 1 1 J r'  J rp C fAJ r'  XJ rpg C XAJ r' D 0: @t 2 4 Put y D J r';

b D J rp:

Then we have @ 1 1 y  b C fAy  Xbg C XAy D 0; @t 2 4

yjt D0 D 0:

The unique solution of the above equation is y D A1 Xb: Now the equation for ' is written as @ i 1 1 '  p C < rp; y >  tr.AX / C < y; Hp y >D 0; @t 2 8 8

'jt D0 D 0:

So we have 1 @ i i '  p  < Jb; A1 Xb >  tr.AX / C < A1 Xb; JXb > D 0: @t 2 8 8 Using the antisymmetry < JAu; v > D  < J u; Av >; we have < Jh.A/u; v > D  < Ju; h.A/v > if h.x/ is an odd function; < Jh.A/u; v > D < Ju; h.A/v > if h.x/ is an even function: If h.x/ is an even function, we have < Jh.A/u; u > D < Ju; h.A/u > D < J 2 u; J h.A/u > D  < J h.A/u; u>:

386

15 The Pseudo-Differential Operator Technique

So we have < Jb; A1 Xb > D 0; < A1 Xb; JXb > D  < A1 X 2 b; Jb > D < JA1 X 2 b; b>: ' is obtained by the following formula: 1 ' D pt C 8

Z 0

t

i tr.AX /ds  8

Z

t

< JA1 X 2 b; b > ds:

0

The integrals can be computed as 1 8

Z

Z 1 t tanh2 .As=2/ds 2 0 Z  1 t 1 D ds 1 2 0 cosh2 .As=2/  t 1  .At=2/1 tanh.At=2/ D 2

t

X 2 ds D 0

and 1 8

Z 0

t

Z 1 t AX ds D A=2 tanh.As=2/ds 2 0 it 1h logfcosh.As=2/g D 0 2 1 D logfcosh.At=2/g: 2

Then we have 1 ' D pt C trŒlogfcosh.At=2/g  2 1 D pt C trŒlogfcosh.At=2/g  2

  it < JA1 1  .At=2/1 tanh.At=2/ b; b > 2 it2 < G.At=2/J rp; rp > : 4

The main part of ' is     tr it2 At pm t C pm1 t C log cosh  < G.At=2/J rpm ; rpm > 2 2 4 with A D iJHpm . Step 2: The estimation of the main part We note first that all the eigenvalues of A are real. If A has a nonzero eigenvalue , then  is also an eigenvalue of A. These facts are shown in the following. Set

15.4 Fundamental Solution by Symbolic Calculus: The Degenerate Case

387

..u; v// D < iJ u; vN >: Then we have

N D ..u; Av//; ..Au; v// D ..u; Av//

and ..Au; u// D ..Hp u; u//  0: Assume Au D u, with  ¤ 0. Then we have 0 < ..Hp u; u// D ..Au; u// D ..u; u//: This means that ..u; u// ¤ 0. On the other hands the following holds: N ..Au; u// D ..u; Au// D ..u; u// D ..u; u//; N We also have and hence we have  D . ANu D Nu; and then  is an eigenvalue of A. By Remark 15.4.2, we have the estimation ˇ ˇ ˇ ˇ 1 C ˇ ˇ ˇp ˇ  C e tr At =2 ; ˇ detfcosh.At=2/g ˇ and by the assumptions on the characteristic set, we have '.x; /  C <  >m t '.x; /  c <  >m1 t

if pm .x; /  cjjm ; in ˙:

We need precise arguments to obtain an estimate for ' near the set ˙. According to [72], we have 0 e ' 2 S1=2;1=2 : Step 3: The construction of the asymptotic term We shall give an idea of the method of construction of ej .tI x; /. The main difficulty of the construction appears on the characteristic set †. I this step we study how to construct ej .tI x; / near †. For the precise proof, see Iwasaki [72]. Let     it2 At tr log cosh  < G.At=2/J rp; rp >; '.tI x; / D pt C 2 2 4

388

15 The Pseudo-Differential Operator Technique

where

G.x/ D .1  x 1 tanh x/=x:

Proposition 15.4.3. We have @ ' X  1 j j .p; e ' / e C @t 2i 2

j D0

i D < .1  tanh.At=2//J rp; 2

i i > C tr.AY /  < ; AJ 8 8

>;

where      it2 At 1  ; (15.4.12) D r tr log cosh 2 2 4 Y D 2r.A1 tanh.At=2//J rp C J r :

(15.4.13)

Proof. By the definition of ', (15.4.12) and (15.4.13), we have  At J rp C J ; J r' D 2A1 tanh 2  At JH ' D 2i tanh C Y: 2

(15.4.14) (15.4.15)

Equations (15.4.14) and (15.4.15) imply the following identities: 

i 1 < J rp; r' > D i < A1 tanh.At=2/J rp; rp >  < rp; J > 2i 2 i (15.4.16) D < J rp; >; 2

where we used that A1 tanh.At=2/ is an even function of A. Then 1 1 i  tr.JH p JH ' / D tr.A tanh.At=2/=2/ C tr.AY / 8 2 8

(15.4.17)

and 

1 < J r'; Hp J r' > 8 1 D  < A1 tanh.At=2/J rp; Hp A1 tanh.At=2/J rp > 2 1 1  < A1 tanh.At=2/J rp; Hp J >  < J ; Hp J 2 8

>

15.4 Fundamental Solution by Symbolic Calculus: The Degenerate Case

i < A1 tanh2 .At=2/J rp; rp > 2 i i  < tanh.At=2/J rp; >  < ; AJ >: 2 8

389

D

(15.4.18)

On the other hand, we have 1 tr.A tanh.At=2/=2/ 2   1 0 < G.At=2/ C .At=2/G .At=2/ J rp; rp > 2 1 D p C tr.A tanh.At=2/=2/ 2 i (15.4.19)  < A1 tanh2 .At=2/J rp; rp >; 2

@ ' DpC @t it  2

where we used

x 0 x tanh2 x G .x/ D : 2 2 x By (15.4.16)–(15.4.19), we obtain the assertion. G.x/ C

Now fix an integer N  3. Since m ; p 2 S1;0

we have

0 e ' 2 S1=2;1=2 ;

 1 j m j2 j .p; e ' / 2 S1=2;1=2 2i

and N  X 1 j m 3 2 j .p; e ' / 2 S1=2;1=2 : 2i

j D3

Then e0 .tI x; / D e

'

satisfies the equation

@ ' X  1 j j .p; e ' / D g0 e ' ; e C @t 2i N

j D0

with

m 3

2 g0 e ' 2 S1=2;1=2 ;



390

15 The Pseudo-Differential Operator Technique

because g0 D

i < .1  tanh.At=2//J rp; 2 N  X 1 j C j .p; e ' /e ' : 2i

i i > C tr.AY /  < ; AJ 8 8

> (15.4.20)

j D3

Note that g0 is a polynomial of J rp whose coefficients are represented by functions of A and its derivatives are given by (15.4.12) and (15.4.13). For any smooth function h.x; /, we have X  1 j @ j .p; he ' / .he ' / C @t 2i N

j D0

N  X @ 1 j .j .p; he ' /  hj .p; e ' // h e ' C D @t 2i j D1 9 8 N = e ' ; 2 .p; he ' /  h2 .p; e ' / D 2 .p; h/e '  < J rh; Hp J r' > e '   1 D  tr.JHp JHh /  < J rh; Hp J r' > e ' : 2 Then we have 2  X 1 j .j .p; he ' /  hj .p; e ' // 2i

j D1

D

1 < J rp; rh > e ' 2i

15.4 Fundamental Solution by Symbolic Calculus: The Degenerate Case

391

 1 2  1  C  tr.JHp JHh /e '  < J rh; Hp J r' > e ' 2i 2 D .Qh/e ' ; where Q is a second-order partial differential operator defined by  1 2  1 1 Qh D  tr.JHp JHh /  < J rh; Hp J r' > : < J rp; rh > C 2i 2i 2 So we have N  X @ 1 j j .p; he ' / .he ' / C @t 2i j D0  X  1 j @ ' h C hg0 C Qh e C D .j .p; he ' /  hj .p; e ' //: @t 2i 3j N

Next we shall approximate Q. Using J r' D 2A1 tanh.At=2/J rp C J ; we have 1 1 < iJ rp; rh > C < iJHp A1 tanh.At=2/iJ rp; rh > 2 2 i 1  tr.AJHh / C < AJ ; i rh > 8 4 1 1 i D < ftanh.At=2/  1giJ rp; rh > C tr.AJHh / C < AJ ; i rh > 2 8 4 1 D Q0 h C < AJ ; i rh >; 4

Qh D 

where

i 1 < ftanh.At=2/  1giJ rp; rh >  tr.AJHh /: 2 8 If f0 D 1, we formally have Q0 h D

1 X

@ @t

! fk e

kD0

D

'

N  X 1 j j C 2i j D0

p;

1 X

! fk e

'

kD0

1 1 1 1  X X X X @ 1 fk C < AJ ; i rfk > e ' fk g0 C Q0 fk e ' ; C @t 4 kD1 1 X

C

kD0

kD1

kD1

X  1 j .j .p; fk e ' /  fk j .p; e ' //: 2i

kD1 3j N

392

15 The Pseudo-Differential Operator Technique

Put n` .f1 ; f2 ; : : : ; f` / D

1 < AJ ; i rf` > C 4

X j Ck D`C3 1  k  `, 3  j  N

 1 j    j .p; fk e ' /  fk j .p; e ' / e ' : 2i Then we have  X  1 j  X  @ X fk e ' C j p; fk e ' @t 2i 1

D

kD0 1 X

kD1

1

N

j D0

@ fk C @t

1 X

kD0

fk g0 C

kD0

1 X

1  X ' Q0 fk e C n` .f1 ; f2 ; : : : ; f` /e ' :

kD1

`D1

It is natural to try to find solutions fk for the following equations: f1 is the solution of @ f1 C Q0 f1 C g0 D 0; @t

f1 jt D0 D 0:

(15.4.21)

fk , with k  2, is the solution of @ fk C fk1 g0 C Q0 fk C nk1 .f1 ; : : : ; fk1 / D 0; @t

fk jt D0 D 0: (15.4.22)

However, we cannot solve the aforementioned equations exactly. We can only find the approximate solution of the aforementioned equations. We prepare the ground for solving (15.4.21) by the following proposition. Let K.t/ be a smooth function valued in an `  ` matrix. Set .t; s) be the unique solution of d dt

C t K.t/



t

.t; s/ D 0;

(15.4.23)

.s; s/ D 0:

Proposition 15.4.4. Let g.t; / be a homogeneous polynomial of degree k in the variable  D t .1 ; 2 ; : : : ; ` /, with coefficients smooth functions of t. Then the following polynomial defined by Z

t

h.t; / D

g.s; .t; s//ds 0

(15.4.24)

15.4 Fundamental Solution by Symbolic Calculus: The Degenerate Case

393

is also a homogeneous polynomial of degree k with respect to , and satisfies @ @ h.t; /C < K.t/; h.t; / > D g.t; /; @t @ h.0; / D 0;   @ where @

h.t; / D t @ @1 h.t; /; @ @2 h.t; /; : : : ; @ @ h.t; / : `

Proof. Set G.t; / D

@ g.t; /. @

Then by (15.4.23), we have

@ h.t; / D g.t; / C @t

Z

t

d .t; s/; G.s; .t; s// > ds dt

< Z

0 t

D g.t; / 

< .t; s/K.t/; G.s; .t; s// > ds Z t t D g.t; /  < K.t/; .t; s/G.s; .t; s//ds > 0

0

@ D g.t; /  < K.t/; h.t; / >; @ because

@ h.t; / D @

Z

t t

.t; s/G.s; .t; s//ds

0



holds by (15.4.24). Corollary 15.4.5. For an `  ` constant matrix B, set .t; s/ D .1 C e Bs /.1 C e Bt /1 and

Z

t

h.t; / D

g.s; .t; s//ds: 0

Then we have @ 1 @ h.t; / C < B.tanh.Bt=2/  1/; h.t; / > D g.t; /; @t 2 @ h.0; / D 0:

Proof. Since in our case we have K.t/ D it suffices to show

1 B.tanh.Bt=2/  1/; 2 

Z

t

.t; s/ D exp  s

 K.r/dr :

394

15 The Pseudo-Differential Operator Technique

Since it is easy to see that Z

t 1 K.r/dr D B.t  s/  logŒcosh.Bt=2/ C logŒcosh.Bs=2/;  2 s  Z t  t  s  exp  K.r/dr D exp B cosh.Bs=2/.cosh.Bt=2//1 2 s

D .1 C e Bs /.1 C e Bt /1 ; 

we get the desired assertion. Next we shall go back to solve (15.4.21).

Lemma 15.4.6. We can obtain an approximate solution of (15.4.21), in the sense @ f1 C Q0 f1 C g0 D G1 ; @t

f1 jt D0 D 0;

where G1 is a polynomial of b.D iJ rp/ with order less than g0 , in the sense that m2 : G1 e ' 2 S1=2;1=2 1=2

Moreover, e1 .tI x; / D f1 .t; x; ; b/e ' 2 S1=2;1=2 . Proof. Set b D iJ rp. We will prove the assertion by induction over the degree of g0 with respect to b. Assume that g0 .t; x; ; b/ is a homogenous polynomial of degree k with respect to b. Then Z f1 .t; x; ; b/ D 

t

g0 .s; x; ; .t; s/b/ ds 0

is a homogeneous polynomial of degree k and satisfies   @ r f1 .t; x; ; b/ D .rf1 /.t; x; ; b/ Ct A f1 .t; x; ; b/; @b where

.t; s/ D .1 C e As /.1 C e At /1 :

Using Corollary 15.4.5 yields   1 @ f1 .t; x; ; b/ C < .tanh.At=2/  1/b; r f1 .t; x; ; b/ > @t 2 1 D g0 C < .tanh.At=2/  1/b; .rf1 /.t; x; ; b/ >; 2 f1 .0; x; ; b/ D 0;

15.4 Fundamental Solution by Symbolic Calculus: The Degenerate Case

395

1=2

and we can show that f1 .t; x; ; b/e ' 2 S1=2:1=2 . We have the following fact:  @2 t tr.AJHf1 / D tr AJ A 2 f1 .t; x; ; b/A C GQ 1 @b m2 with GQ 1 e ' 2 S1=2:1=2 . Then we have

 i Q i @2 @ t f1 C Q0 f1 C g0 D  G1  tr AJ A 2 f1 .t; x; ; b/A @t 8 8 @b 1 C < .tanh.At=2/  1/b; .rf1 /.t; x; ; b/>: 2 m2 We can prove that h.tanh.At=2/  1/b; .rf1 /.t; x; ; b/ie ' belongs to S1=2;1=2 , 2

@ and that tr.AJ t A @b 2 f1 .t; x; ; b/A/ is a homogenous polynomial of degree k  2 with respect to b. This leads to the desired conclusion. 

Instead of (15.4.22) for fk .k  2/, we need to solve the following equations step by step: @ fk C Q0 fk C fk1 g0 C nk1 .f1 ; : : : ; fk1 / C Gk1 D Gk ; @t

fk jt D0 D 0; (15.4.25)

under the condition m3=2k=2 Gk e ' 2 S1=2;1=2 :

We can obtain the solution fk of (15.4.25) and also show that ej .tI x; / D j=2 by a similar method to the construction of f1 . fj .t; x; ; b/e ' 2 S1=2;1=2 Step 4: The construction of the fundamental solution We have constructed ej .tI x; /, for j D 1; 2; : : : ; N  1, such that 0 1 NX  1 @ w ejw .tI x; D/A D rN .tI x; D/; C p w .x; D/ @ @t j D0

mN=2 . So we can construct the symbol of the fundamental with rN .tI x; / 2 S1=2;1=2 solution of the form

e.t/ D

N 1 X

Z ej .t/ C

1 t N X

ej .t  s/ ıw

.s/ds;

0 j D0

j D0

mN=2 if m  N=2  0. In fact, we can construct .t/ as a unique with .t/ 2 S1=2;1=2 solution of the following equation by the similar method as in Theorem 15.2.1:

Z rN .t/ C

.t/ C

t

rN .t  s/ ıw 0

.s/ds D 0:

396

15 The Pseudo-Differential Operator Technique

In this case we apply the estimate of symbols of the multi-product of the pseudomN=2 differential operators of Weyl symbols. Then we have the solution .t/ 2 S1=2;1=2 P 1 of the previous integral equation. So e.t/ has the expansion e.t/  jND0 ej .t/ 2

mN=2 j=2 S1=2;1=2 for any N . Since each ej .t/ belongs to S1=2;1=2 , we can show that e.t/  PN 1 N=2  j D0 ej .t/ 2 S1=2;1=2 . This completes the proof.

15.4.2 The Symbol pm D

1 2

P`

j D1

qj2

If the principal symbol pm has the exactly the double characteristic †, that is, P pm .x; / D 12 `j D1 qj2 .x; /, then the following theorem provides a closed-form expression for '. P Theorem 15.4.7. If pm .x; / D 12 `j D1 qj2 .x; /, then we can choose i t 1 h ' D pm1 t C tr logfcosh.M t=2/g C < F .M t=2/q; q >; 2 2 where

tanh x ; q D t .q1 ; : : : ; q` /; x and M is an `  ` Hermitian matrix defined by F .x/ D

M D .Mjk /;

Mjk D i < rqj ; J rqk >:

We note that if the qj are symbols of vector fields, then the Mjk are symbols for the corresponding commutators. Proof. The previous formula can be obtained in two ways. One is to repeat the proof under the assumption rp D

` X

qj rqj ;

Hp D

j D1

` X

rqjt .rqj /:

j D1

The other way is to use the result obtained in Theorem 15.4.1: ` it2 t X < G.At=2/.iJ rqj t=2/qj ; qk rqk > < G.At=2/J rp; rp > D 4 2 j;kD1

D

` t X qj qk .FQ .Mt=2//jk ; 2 j;kD1

15.4 Fundamental Solution by Symbolic Calculus: The Degenerate Case

397

where FQ .x/ D xG.x/ D 1  F .x/. Then the following identity holds: pt 

` it2 t 1X 2 qj t  < FQ .M t=2/q; q > < G.At=2/J rp; rp > D 4 2 2 j D1

t D < F .M t=2/q; q>: 2 On the other hand, we have AS D SM; where S is a .2n  `/ matrix defined by S D .iJ rq1 ; iJ rq2 ; : : : ; iJ rq` /: So we get detfcosh.At=2/g D detfcosh.M t=2/g: This completes the proof.



15.4.3 The Special Case of Quadratic Symbols Let p.x; / be a polynomial in .x; / of degree at most 2. In this case the fundamental solution E.t/ is obtained   as a pseudo-differential operator of Weyl symbol e.tI x; / D exp  '.t; x; / , where '.tI x; / D pt C

    it2 At tr log cosh  < G.At=2/J rp; rp >; 2 2 4

because '.tI x; / is the exact solution of the equation 

1 d' 1 1 C p  1 .p; '/ C 2 .p; '/  < J r'; Hp J r' >D 0: dt 2i 4 8

More precisely, we have the following result. This is the key theorem for the construction of the fundamental solution for the heat equation with polynomial coefficients. It is also useful for the construction of the fundamental solution of degenerate parabolic equations. Theorem 15.4.8. If p.x; / is a quadratic polynomial with respect to X D t.x; / 2 Rd  Rd , 1 p D hX; HX i C i hX; p0 i C b; 2

398

15 The Pseudo-Differential Operator Technique

then E.t/ D e w .tI x; D/ is given by  n e bt e.tI x; / D p exp i hJ tanh.At=2/X; X i det cosh.At=2/ C thJ tanh.At=2/.At=2/

1

o t2 X; Jp0 i C hJ G.At=2/Jp0 ; Jp0 i ; 4

where A D iJH is a 2d  2d matrix,  J D and

0 I I 0

;

G.x/ D .1  x 1 tanh x/=x:

Proof. Since rp D HX C ip0 ;

Hp D H;

we have hG.At=2/Jp; pi D hG.At=2/JHX; HX i C hG.At=2/JHX; p0 i C i hG.At=2/Jp0 ; HX i  hG.At=2/Jp0 ; p0 i D hJ G.At=2/JHX; JHX i C i hJ G.At=2/JHX; Jp0 i C i hJ G.At=2/Jp0 ; JHX i  hJ G.At=2/Jp0 ; Jp0 i; and hence t2 hG.At=2/Jp; pi D hJ G.At=2/.At=2/X; .At=2/X i 4 t C hJ G.At=2/.At=2/X; Jp0i 2 t t2 C hJ G.At=2/Jp0 ; .At=2/X i  hJ G.At=2/Jp0 ; Jp0 i: 2 4 Using that t

AJ D JA

and t

.J G.At=2// Dt G.At=2/t J D t G.At=2/J D J G.At=2/;

for the odd function G.x/, we have t2 hG.At=2/Jp; pi D hJ G.At=2/.At=2/2 X; X i C thJ G.At=2/.At=2/X; Jp0i 4 t2  hJ G.At=2/Jp0 ; Jp0 i: 4

15.4 Fundamental Solution by Symbolic Calculus: The Degenerate Case

Since x 2 G.x/ D x  tanh x;

399

xG.x/ D 1  x 1 tanh x;

it holds that hJ G.At=2/.At=2/2 X; X i D hJ.At=2  tanh.At=2//X; Xi it D  hHX; X i  hJ tanh.At=2/X; X i 2 and hJ G.At=2/.At=2/X; Jp0i D hJX; Jp0 i  hJ.At=2/1 tanh.At=2/X; Jp0 i D hX; p0 i  hJ.At=2/1 tanh.At=2/X; Jp0i: Therefore, we have i

t t2 hG.At=2/Jp; pi D hHX; X i  i hJ tanh.At=2/X; X i 4 2 C ithX; p0 i  ithJ.At=2/1 tanh.At=2/X; Jp0 i t2 i hJ G.At=2/Jp0 ; Jp0 i 4 D pt  bt  i hJ tanh.At=2/X; X i  ithJ.At=2/1 tanh.At=2/X; Jp0 i t2  i hJ G.At=2/Jp0 ; Jp0 i: 4

Finally, we have    tr At Q D bt C log cosh. / C i hJ tanh.At=2/X; X i 2 2 C ithJ.At=2/1 tanh.At=2/X; Jp0 i t2 C i hJ G.At=2/Jp0 ; Jp0 i: 4 Noting that     tr At 1 ; exp  log cosh. / Dp 2 2 det cosh.At=2/ we get the assertion.



400

15 The Pseudo-Differential Operator Technique

15.4.4 A Key Theorem for Eigenfunction Expansion In a forthcoming section we shall show a method of obtaining eigenfunctions using the fundamental solution represented by pseudo-differential operators of Weyl symbols. The following result is the key of a future proof of the expansion for the kernel of the fundamental solution obtained as a pseudo-differential operator. Proposition 15.4.9. If the symbol of a pseudo-differential operator is of the form p.x; / D h.x; /g.x; /, then the kernel of operator p w .x; D/ in Rd is given by .2/d

Z

0

Rd

e i.xx / p

where gQ

 x C x0 2

r 2

 r @   r ˇˇ ; ;  d D h ; i ;q ˇ gQ rDxCx 0 ;qDxx 0 2 @q 2

Z  ; q D .2/d

Rd

e i q g

r

Proof. If the symbol has the expansion h.x; / D x C x 0 and q D x  x 0 , we formally have Z .2/d

0

e i.xx / p

 x C x0

Z  ;  d  D .2/d

 ;  d:

2 P

X

˛

a˛

2 Rd ˛ Z   X r  @ ˛ a˛ .2/d e i q g D i ;  d 2 @q 2 Rd ˛ r @   r ˇˇ : ; i ;q ˇ Dh gQ rDxCx 0 ;qDxx 0 2 @q 2 Rd

a˛ .x/ ˛ , then with r D r  r  ˛ e i q g ;  d 2 2

 r 



15.5 The Hermite Operator In the rest of this section we will provide symbols of the fundamental solution as pseudo-differential operators according to Theorem 15.4.8 for several examples that have been studied in previous chapters by different methods. First we will study the heat equation corresponding to the Hermite operator. We will also provide its eigenfunction expansion according to Proposition 15.4.9. The one-dimensional Hermite operator considered in this section is given by P D where c is a positive constant.

 1 @2  2 C c2 x2 ; 2 @x

(15.5.26)

15.5 The Hermite Operator

401

15.5.1 Exact Form of the Symbol of the Fundamental Solution The next result deals with the fundamental solution E.t/ of the operator (15.5.26) as a pseudo-differential operator. We note that its kernel K.tI x; x 0 / is given by 1 K.tI x; x / D 2 0

Z

1

e

i.xx 0 /

1

 x C x0 e tI ;  d : 2

Theorem 15.5.1. The fundamental solution is E.t/ D e w .tI x; D/, with the Weyl symbol  1 tanh.ct=2/ 2 2 e.tI x; / D exp  .c x C  2 / : cosh.ct=2/ c The kernel K.tI x; x 0 / is given by 0

r

K.tI x; x / D

 n o c c 2 02 0 .x C x / cosh.ct/  2xx : exp  2 sinh.ct/ 2 sinh.ct/

Proof. The symbol of the operator (15.5.26) is given by 1 2 . C c 2 x 2 /: 2

.P / D Then choosing in Theorem 15.4.8

 H D

c2 0



0 1

and X Dt .x; /; we have 1 e.tI x; / D p expfi hJ tanh.At=2/X; X ig: det cosh.At=2/ If c D 0, then e.tI x; / D e jj

2 t =2

:

Assume that c ¤ 0. The 2  2 matrix A D iJH has eigenvalues c; c and eigenvectors U

C

 D

1 ic

;

U



 D

1 ic

:

402

15 The Pseudo-Differential Operator Technique

It is easy to see that

X D c1 U C C c2 U  ;

with c1 D

x  Ci ; 2 2c

c2 D

x  i : 2 2c

Then we have iJ tanh.At=2/X D i tanh.ct=2/.c1 J U C  c2 J U  / D tanh.ct=2/W; where

  W Dt cx; : c

We have 1 2 . C c 2 x 2 /; c p det cosh.At=2/ D cosh.ct=2/; tanh.ct=2/ 2 . C c 2 x 2 /: i hJ tanh.At=2/X; X i D c hW; X i D

Thus the kernel of the fundamental solution is given by  Z 1 x C x0 i.xx 0 / e e tI K.tI z; s; z ; s / D ;  d 2 R 2 Z 1 1 0 e i.xx / D 2 R cosh.ct=2/   tanh.ct=2/ 1 2 c .x C x 0 /2 C  2 d   exp  c 4 r c 1 D p 2  cosh.ct=2/ tanh.ct=2/   c .x  x 0 /2 0 2 tanh.ct=2/.x C x / C :  exp  4 tanh.ct=2/ 0

0

Using the equalities 1 2 D ; tanh.ct=2/ tanh.ct/ 1 2 tanh.ct=2/  D ; tanh.ct=2/ sinh.ct/ 1 sinh.ct=2/ cosh.ct=2/ D sinh.ct/ 2

tanh.ct=2/ C

15.5 The Hermite Operator

403

yields the following expression for the heat kernel: 0

r

K.tI x; x / D

 n o c c 2 02 0 .x C x / cosh.ct/  2xx : exp  2 sinh.ct/ 2 sinh.ct/ 

15.5.2 Eigenfunction Expansion We shall apply Proposition 15.4.9 to e.tI x; / given by the previous theorem. The Weyl symbol in the case of the Hermite operator can be written as e.tI x; / D

 1 tanh.ct=2/ 2 2 exp  .c x C  2 / cosh.ct=2/ c

D h.t; x; /e1 .tI x; /; where e1 .tI x; / D 2e and h.tI x; / D

ct =2

 1 2 2 2 exp  .c x C  / ; c

 n o 1 1 2 2 2 exp x C  / : 1  tanh.ct=2/ .c 1 C e ct c

With the substitution

w D e ct ;

we have    1 2w @  @2 2 2 D c r =4  2 exp h tI r=2; i @q 1Cw c.1 C w/ @q    1 @ @ 2w D cr=2 C cr=2  : exp 1Cw c.1 C w/ @q @q If we let r D x C x 0 and q D x  x 0 , we have cr=2 C and

1 @ D @q 2

@ 1 cr=2  D @q 2



  @ @ C cx C  0 C cx 0 @x @x

   @ @ 0  C cx C C cx : @x @x 0

404

15 The Pseudo-Differential Operator Technique

Then the following equation holds:     @  1 w    h tI r=2; i D Bx C .Bx 0 / exp .Bx / C Bx 0 ; @q 1Cw .1 C w/ where

Note that

 1  @ C cx ; Bx D p 2c @x ŒBx ; .Bx /  D 1;

Set .x/ D

 1  @ 0 C cx Bx 0 D p : 2c @x 0

ŒBx C .Bx 0 / ; .Bx / C Bx 0  D 0:  c  14

 We conclude with the following result.

 cx 2  exp  : 2

(15.5.27)

w .tI x; D/. Then we have Proposition 15.5.2. Let eQ1 .tI x; x 0 / be the kernel of e1

eQt;1 .tI x; x 0 / D

p w .x/ .x 0 /:

Proof. We have the following computation:  Z 1 x C x0 i.xx 0 / e e1 tI ;  d eQ1 .tI x; x / D 2 R 2 p p   2 w c c.x  x 0 /2 c.x C x 0 /2 D exp   2 4 4 p D w .x/ .x 0 /: 0

 next with the following expansion regarding the operator  We shall @ deal  h tI r=2; i @q . Proposition 15.5.3. (1) Let Cj .S / be the coefficients defined by the following expansion: 1 h w i X 1 Cj .S / j exp S D w : 1Cw .1 C w/ jŠ j D0

Then the following recursive formula holds: C`C1 .S / D S C` .S /  .2` C 1/C` .S /  `2 C`1 .S /; C0 .S / D 1:    (2) If S D Bx C .Bx 0 / .Bx / C Bx0 , then Cj .S / .x/ .x 0 / D

j .x/

j .x

0

/;

`  0;

15.5 The Hermite Operator

405

where j .x/

D .B  /j .x/:

Proof. (1) Consider the analytic function f .z/ defined by  z  1 f .z/ D exp S : .1 C z/ 1Cz  .1 C z/f .z/ D exp

Since we have

 z S ; 1Cz

it follows that f 0 .z/.1 C z/2 C .1 C z/f .z/ D f .z/S;

f .0/ D 1:

Differentiating ` times, we obtain .1 C z/2 f .`C1/ .z/ C 2`.1 C z/f .`/ .z/ C `.`  1/f .`1/ .z/ C .1 C z/f .`/ .z/ C `f .`1/ .z/ D f .`/ .z/S: Making z D 0, we obtain the desired equation for C` .S / D f .`/ .0/. (2) We note that the following equations hold: (i) Bx .x/ D 0; (ii)

Bx

` .x/

D`

`1 .x/;

.Bx /k .Bx /k .x/ D kŠ .x/;

(iii) S



j .x/

j .x

0

 / D

j C1 .x/

Cj

2

j C1 .x

j 1 .x/

0

/ C .2j C 1/

j 1 .x

0

j .x/

j .x

0

/

/:

We shall proceed by induction with respect to j . The assertion obviously holds for j D 0, because C0 .S / D 1 for all S . Note that for any operator S; the operators Cj .S / satisfy part (1). Assume that formula (2) holds for all `  j . Then we have       Cj C1 .S / .x/ .x 0 / D S Cj .S / .x/ .x 0 /  .2j C 1/Cj .S / .x/ .x 0 /    j 2 Cj 1 .S / .x/ .x 0 /   D S j .x/ j .x 0 /  .2j C 1/ j .x/ j .x 0 /  j2 D

j 1 .x/ j 1 .x 0 j C1 .x/ j C1 .x /;

0

/

where the last equality is obtained by (iii). Thus the induction method provides the desired conclusion.

406

15 The Pseudo-Differential Operator Technique

We still have to prove formulas (i), (ii), and (iii). We shall next give the sketch of the proof for (i). Bx

D ŒBx ; .Bx /`  .x/ C .Bx /` Bx .x/

` .x/

D `.Bx /`1 .x/ D ` `1 .x/: Formula (ii) is proved by using the following equation inductively: Bxk .Bx /k .x/ D Bxk D D We have   .Bx / C Bx0 .

j .x/ j .x

0

 // D .Bx / D

noting

k .x/ k1 Bx k k1 .x/ kBxk1 .Bx /k1

.Bx /

 j .x/

j C1 .x/

j .x/

.x/:

D

j .x

j .x 0

0

/Cj



/C

j .x/

j .x/

Bx 0

j 1 .x

0

j .x

0

 /

/;

j C1 .x/:

Then we have S



j .x/ j .x

0

/



  D Bx C .Bx 0 / .

j C1 .x/

j .x

0

/Cj

j .x/

0

j 1 .x

0

D .j C 1/ j .x/ j .x / C j C j C1 .x/ j C1 .x 0 / C j j C1 .x/

//

0

D .Bx j C1 .x// j .x / C j.Bx j .x// j 1 .x / C j C1 .x/..Bx 0 / j .x 0 // C j j .x/..Bx 0 /

D

0

2

j 1 .x/ j .x/

0 j C1 .x / C .2j C 1/

j 1 .x 0

j .x j .x/

0

j 1 .x

0

//

/

/

j .x

0

/ C j2

j 1 .x/

j 1 .x

0

/:

The proof of (iii) is complete. Theorem 15.5.4. The kernel K.tI x; x 0 / of e w .tI x; D/ has the following expansion: 1 X 1 ct .j C1=2/ e K.tI x; x / D jŠ 0

j .x/

j .x

j D0

with

oj n 1  @ C cx D p .x/;  @x 2c .x/ defined by (15.5.27). j .x/

with

0

/;

15.6 The Grushin Operator

407

Proof. By Propositions 15.4.9, 15.5.2, and 15.5.3, we have  1 X 1 j @ eQ1 .tI x; x 0 / D w Cj .S /eQ1 .tI x; x 0 / K.tI x; x 0 / D h tI r=2; i @q jŠ j D0

D

1 X j D0

1 X 1 j C1=2 1 j C1=2 Cj .S / .x/ .x 0 / D w w jŠ jŠ

j .x/

j .x

j D0

0

/: 

The above theorem tells us that the Hermite operator has p the1eigenvalues fc.j C and the corresponding eigenfunctions f .x/= j Šgj D0. 1=2/g1 j j D0 These results are useful in the study of the following Grushin operator. Taking the Fourier transform, we obtain symbol similar to that of the Hermite operator.

15.6 The Grushin Operator In this section we shall provide the exact form for the symbol of the fundamental solution of the Grushin operator. This is a degenerate operator on R2 defined as 1 P D 2



2 @2 2 @ C x @x 2 @y 2

:

Theorem 15.6.1. We have E.t/ D e w .tI x; y; Dx ; Dy / with e.tI x; y; ; / D

 1 tanh.t=2/ 2 exp  . C x 2 2 / ; cosh.t=2/ 

and with its kernel given by K.tI x; y; x 0 ; y 0 / D

Z

1 3 2

0 v

e i.yy / t

r

v 2 sinh.v/

2 t  R n o v 2 .x 2 C x 0 / cosh.v/  2xx 0 dv:  exp  2t sinh.v/ Proof. Since we have 1 2 . C x 2 2 /; 2 using Theorem 15.5.1 we easily get the symbol of the fundamental solution. The kernel of the fundamental solution is obtained as in the following:  w .P / D

K.tI x; y; x 0 ; y 0 / D

  1 2 Z  x C x0 y C y 0 0 0 ; ; ;  dd e i.xx /Ci.yy / e tI 2 2 2 R2

408

15 The Pseudo-Differential Operator Technique

Z

r

 2 sinh.t/ R  n o  2 .x 2 C x 0 / cosh.t/  2xx 0 d  exp  2 sinh.t/ Z r 1 v i.yy 0 / tv D e 3 2 sinh.v/ 2 t 2  R n o v 2 .x 2 C x 0 / cosh.v/  2xx 0 dv:  exp  2t sinh.v/

D

1 2

0

e i.yy /



15.7 Exact Form of the Symbol of the Fundamental Solution for the Sub-Laplacian Let x; y; s 2 R, z D x C iy 2 C and consider the following vector fields: ZD

@ @ C i aNz ; @z @s

@ @ ZN D  i az ; @Nz @s

T D

@ : @s

This section is concerned with the sub-Laplacian on R3 defined by P D

 1 N N Z Z C ZZ C i˛T: 2

Theorem 15.7.1. Let a 2 R and ˛ 2 C be constants such that j˛j < a and a > 0. Then E.t/ D e w .tI x; y; s; Dx ; Dy ; Ds / is obtained with e.tI x; y; s; ; ; / D

 tanh.a t/ e ˛ t exp  f. C 2ay/2 C .  2ax/2 g cosh.a t/ 4a

and its kernel is given by K.tI z; s; z0 ; s 0 / D

Z

1 1



 a exp i fs  s 0 C 2a.=.zzN0 //g 2 2 sinh.a t/

a jz  z0 j2 C ˛ t d: tanh.a t/

Proof. We have  w .L/ D

1 f. C 2ay/2 C .  2ax/2 g  ˛; 4

15.7 Exact Form of the Symbol of the Fundamental Solution for the Sub-Laplacian

409

by Theorem 15.4.8, with 0

2a2  2 B 0 H DB @ 0 a

0 2a2  2 a 0

1 a 0 C C 0 A

0 a 1 2

1 2

0

and X D t .x; y; ; /I this yields the symbol e ˛ t expfi hJ tanh.At=2/X; X ig: e.tI x; / D p det cosh.At=2/ Assume that  ¤ 0. The 4  4 matrix A D iJH has the eigenvalues 2a;

2a;

0;

0

and the corresponding eigenvectors 1 1 B i C C DB @ 2i a A; 2a 0

UC

1 1 B i C C DB @ 2i a A; 2a 0

U

1 1 B 0 C C U0 D B @ 0 A; 2a 0

1 0 B 1 C C UQ 0 D B @ 2a A: 0 0

It is easy to see that we have the decomposition X D c0 U0 C cQ0 UQ0 C c1 U C C c2 U  ; with c0 D

x  C ; 2 4a

c1 D

x C iy . C i/ Ci ; 4 8a

cQ0 D

y   ; 2 4a c2 D

x  iy .  i/ i : 4 8a

Then we have iJ tanh.At=2/X D i tanh.a t/.c1 J U C  c2 J U  / D tanh.a t/W; where WD

  ax  ; 2

t

 ay C ; 2

y  C ; 2 4a



x   : C 2 4a

410

15 The Pseudo-Differential Operator Technique

By computation we obtain hW; X i D p

1 f. C 2ay/2 C .  2ax/2 g; 4a

det cosh.At=2/ D cosh.a t/;

i hJ tanh.At=2/X; X i D

tanh.a t/ f. C 2ay/2 C .  2ax/2 g: 4a

Then the kernel of the fundamental solution is given by  1 3 Z 0 0 0 K.tI z; s; z ; s / D e i.ss /Ci.xx /Ci.yy / 2 R3  x C x0 y C y 0 ; ; ; ;  d d d  e tI 2 2  1 3 Z 0 0 0 0 0 e i fss C2a.x yxy /gCi.xx /Ci.yy / D 3 2 R  ˛ t e tanh.a t/  exp  .jj2 C jj2 / ddd cosh.a t/ 4a  Z 1 a D exp i fs  s 0 C 2a.x 0 y  xy 0 /g 2 1 2 sinh.a t/ a  jz  z0 j2 C ˛ t d: tanh.a t/ 0

0



15.8 The Sub-Laplacian on Step-2 Nilpotent Lie Groups We apply the last construction of the fundamental solution to a case of the degenerate operators, that is, to the case of the sub-Laplacian on two-step free nilpotent Lie groups. So let FN CN.N 1/=2 Š RN ˚ RN.N 1/=2 be a connected and simply connected free two-step nilpotent Lie group with the Lie algebra fN CN.N 1/=2 (it is also identified with RN ˚ RN.N 1/=2 ). We fix a basis fXj ; Zj; k j 1  j; k  N; j < kg of the Lie algebra fN CN.N 1/=2 . Their bracket relation is assumed to be ŒXj ; Xk  D 2Zjk

15.8 The Sub-Laplacian on Step-2 Nilpotent Lie Groups

411

for 1  j < k  N; and the group multiplication  W FN CN.N 1/=2  FN CN.N 1/=2 ! FN CN.N 1/=2 is given by RN ˚ RN.N 1/=2  RN ˚ RN.N 1/=2 X X ˝ X  X ˛ 3 xj X j ˚ zjk Zjk ; zQjk Zjk xQ j Xj ˚ X X  X  X xQ j Xj ˚ 7! xj Xj ˚ zjk Zjk  zQjk Zjk X X  xj C xQ j /Xj ˚ D zjk Ce zjk C xj xQ k  xk xQ j Zjk : Let XQj be the left invariant vector fields on FN C.N 1/=2 : d f .g  e tXj /jt D0 XQ j .f /g D dt X X @f @f @f D C xk  xk ; @xj @zkj @zjk k<j

k>j

where g D .x; z/ 2 RN ˚ RN.N 1/=2 Š FN CN.N 1/=2 . Let .x; zI ; / D .xj ; zjk I j ; jk / 2 T  .FN CN.N 1/=2 / Š RN ˚RN.N 1/=2  RN ˚ RN.N 1/=2 be the dual coordinates on the cotangent bundle; then we understand the symbol of vector fields XQ j and their Weyl symbol as 0 1 X X p  w .XQ j / D .XQj / D 1 @j C xk kj  xk jk A p D 1.  ./x/j ;

k<j

k>j

where D ./ is an N  N skew-symmetric matrix defined by   ./

jk

D jk

.1  j < k  N /:

Let P be the sub-Laplacian P D

N 1 X Q2 Xj I 2 j D1

then its Weyl symbol is given by  w .P / D 

N 1X w Q 2  .X j / 2 j D1

D

1 < X; HX >; X D t .x; /; 2

412

15 The Pseudo-Differential Operator Technique

with a 2N  2N matrix H defined by 0

. .//2 ./

H D@

 ./

1 A:

I

We consider the pseudo-differential operator p w .x; z; Dx ; Dz / with the Weyl symbol  w .P / and construct the following fundamental solution E.t/ as a pseudodifferential operator of the Weyl symbol e.tI x; z; ; /: d w e .tI x; z; Dx ; Dz / C p w .x; z; Dx ; Dz /e w .tI x; z; Dx ; Dz / D 0 dt in .0; T /  RN CN .N 1/=2 ; e w .0I x; z; Dx ; Dz / D I: Theorem 15.8.1. The Weyl symbol e.tI x; z; ; / is obtained as follows:  

1 t tanh.i t 0 / e.tI x; z; ; / D p exp  H X; X ; 2 i t 0 det cosh.it .// where

0 0 D @

./

0

0

./

1 A:

Then E.t/u.x; z/ D e w .tI x; z; Dx ; Dz /u.x; z/ Z Z N N.N 1/=2 D .2/ RN RN

RN.N 1/=2 RN.N 1/=2

Q e i <xx;>Ci 

 e.tI .x C x/=2; Q .z C zQ/=2; ; / u.x; Q zQ/ d dQz d dx: Q Corollary 15.8.2. The kernel function of the above fundamental solution K.tI x; z; x; Q zQ/ is given by Z Q Ci =t e i <x; . /x>=t K.tI x; z; x; Q zQ/ D .2 t/N=2N.N 1/=2 RN.N 1/=2   1 i ./  exp  < x  x; Q .x  x/ Q > 2t tanh.i .// s h i ./ i  det d ; sinh.i .// where < z;  > D

P 1j Ci

 e.tI .x C x/=2; Q .z C zQ/=2; ; / d d: The following equation is clear for a symmetric nonsingular matrix M : Z RN

exp. < M ;  > Ci < a;  >/ d

D .det M /1=2 ./N=2 exp. < a; M 1a > =4/: Applying the above equation for M D

t tanh.it . // 2 it . /

and

a D .x  x/ Q  tanh.it .//.x C x/=2; Q we get the assertion of Corollary 15.8.2.

15.9 The Kolmogorov Operator

415

15.9 The Kolmogorov Operator Let P be an operator on R2 defined by P D

@ 1 @2 Cx : 2 2 @x @y

(15.9.31)

The next result deals with the exact form of a symbol of the fundamental solution of the Kolmogorov operator. Theorem 15.9.1. The fundamental solution for the operator (15.9.31) is E.t/ D e w .tI x; y; Dx ; Dy /, with the symbol  t 2 t3 2 e.tI x; y; ; / D exp    ixt   : 2 24 The heat kernel of (15.9.31) is given by "  2#  1 3 3 0 2 0 1 0 exp  .x  x /  2.y  y /  .x C x / : K.tI x; y; x ; y / D t2 2t 2t t p

0

0

Proof. We have

In this case

 H D

0 0

 w .P / D

1 2  C ix: 2

0 ; 1

  ; p0 D 0

and X Dt .x; /: Note that A2 D 0: So we have tanh.At=2/ D At=2; 1 G.At=2/ D At 6 because G.x/ D

1 x C O.x 5 /: 3

bD0

416

15 The Pseudo-Differential Operator Technique

By Theorem 15.4.8, we have e.tI x; y; ; /  n o t2 D exp i hJ.At=2/X; X i C thJX; Jp0 i C hJ.At=6/Jp0 ; Jp0 i 4  3 t t D exp  hHX; X i  i thX; p0 i  hHJp0 ; Jp0 i 2 24  t 2 t3 2 D exp    ixt   : 2 24 The kernel of the fundamental solution is given by K.tI x; y; x 0 ; y 0 /   1 2 Z x C x0 0 0 e i.xx /Ci.yy / e tI ; ;  d d D 2 2 R2 2 (r 3 )2 Z  1 2 0 3 0 2 t .x  x i.x  x t / / 5d d exp 4  2 C iˇ   p D 2 2 24 2t 2t R2  3   1 2 p2 .xx0 /2 Z t 2  2t D p e exp   C iˇ d; 2 24 t R with ˇ D y  y0 

x C x0 t: 2

So we have p n .x  x 0 /2 6ˇ 2 o 2 p 24 exp  K.tI x; y; x ; y / D  3 4 t 2 " 2t t p   2# 3 1 3 1 2.y  y 0 /  .x C x 0 / : D exp  .x  x 0 /2  t2 2t 2t t 0

0

 To conclude, the method of pseudo-differential operators is feasible in several important cases, including degenerate operators (the Kolmogorov case), operators with potential (the Hermite case), and sub-elliptic Laplacians on two-step nilpotent Lie groups.

Appendix

Relationship Between the Heat and Wave Kernels This appendix presents the relationship between the wave kernel and the heat kernel of an operator. Since finding wave kernels is more difficult than finding heat kernels, this is not always an efficient method for computing heat kernels. On the other hand, the number of examples which can be worked out explicitly is limited. Here we provide just the example of the operator @2x . A similar method is used in [62] to provide the explicit formula of the heat kernel on the hyperbolic space. Let L be a second-order operator defined on C01 .Rn / with a self-adjoint nonpositive definite extension on L2 .Rn /. One defines the wave kernel of L to be the distribution u on Rn  .1; 1/ which satisfies the following Cauchy problem: @2t u D Lu; ujt D0 D 0; @t ujt D0 D ı0 ; where ı0 is the Dirac distribution centered at x D 0. The solution u.x; t/ is a wave which is flat at t D 0 and has the initial velocity given by an impulse function centered at the origin. A formal computation shows that p  1 u.x; t/ D p (A.1) sin t L ; L where L is assumed to be positive definite. On the other side, the heat kernel of L is the distribution v on Rn  .0; 1/ such that @t v D Lv; vjt D0 D ı0 : One may formally write v.x; t/ D e tL ;

t > 0:

(A.2)

417

418

Appendix

In order to establish relationship between formulas (A.1) and (A.2), we consider the following Fourier transform identity, where  can be either a number or an operator: 2

e t  D p D p Substituting

p

Z

1

4 t Z 1 4 t

e

2

 s4t

s2

e  4t

1

Z

s2

ds D p e  4t cos.s/ ds 4 t   1 sin.s/ ds: @s 

e

is

L for  yields e

tL

1

D p 4 t

Z e R

2

 s4t

 @s

p

1 L

p sin.s L/

 ds:

To conclude, we have Theorem 15.9.2. Let v.x; t/ be the heat kernel and u.x; t/ the wave kernel for L. Then Z   1 s2 t > 0: (A.3) e  4t @s u.x; s/ ds; v.x; t/ D p 4 t R As an immediate application of this formula, one may consider the operator L D @2x with the wave kernel u.x; t/ D H.t  jxj/, where H.u/ D  1; if u > 0 is the Heaviside function. In this case the derivative in the dis0; if u  0     tribution sense @s u.x; s/ D @s H.s  jxj/ D ı.s  jxj/ yields the Dirac distribution. Then formula (A.3) provides the well-known expression for the heat kernel Z jxj2 1 1 s2 v.x; t/ D p e  4t ı.s  jxj/ ds D p t > 0: e  4t ; 4 t R 4 t Other computations of heat kernels starting from the associated wave kernel constitute a launching ground for future research. A good reference for wave kernels is the paper [59].

Conclusions

At the end of this journey through several techniques of finding explicit formulas for heat kernels, we shall conclude with a few final remarks. Some of the methods presented in this monograph cover only certain operators in most of the cases and are not well suited for the others. We shall discuss in the following the experience gained from using these methods. More precisely, we shall be concerned with what method is suited to which operator. Elliptic and sub-elliptic operators via the geometric method. The heat kernel for a given elliptic operator has a determined physical meaning; i.e., it is the amount of heat transferred between two given points on a Riemannian manifold in a given time. It is physically reasonable to assume that the diffusion of heat occurs along the geodesics of the Riemannian geometry associated with the elliptic operator considered initially. There are a few classes of complete manifolds where the connectivity by geodesics holds; i.e., there is only one geodesic joining any two given points on the manifold. For instance, Hadamard manifolds (negatively curved Riemannian manifolds) and compact manifolds are just a couple of examples. However, the local connectivity by geodesics property always holds on any manifold (Whitehead’s theorem). For the elliptic operators associated with these types of Riemannian geometries, the expression of the heat kernel can be obtained by a closed-form formula that involves a product between the diffusion coefficient, called the volume function, and an exponential term of the action function along the geodesic. This action function, in the case of Riemannian geometry, is obtained by dividing the square of the Riemannian distance by 2t, where t denotes the time parameter. When the geodesics have conjugate points, the expression of the heat kernel is more complicated, as in the case of the circle S 1 . In other cases there might not be any explicit formulas, as the case of the sphere S 2 . We have extended this method to elliptic operators with potential. For these operators the geodesics are replaced by the solutions of the Euler–Lagrange equations associated with the Lagrangian obtained as a difference between the kinetic energy induced by the metric and the potential function. Explicit formulas exist only in the cases when the potential is linear or quadratic. Higher powers lead to unsolved problems, such as the exact solution for the quartic oscillator problem.

419

420

Conclusions

If the operator is sub-elliptic, the appropriate geometry is sub-Riemannian. Unfortunately, in this case the problem of connectivity by geodesics is a very complicated one, since the “local” and “global” behaviors in this case coincide. Since the sub-elliptic operators have always at least one missing direction, it was proved successful in several cases to integrate along the characteristic variety (the space where the principal symbol vanishes) a product between a volume function and the exponential of a modified complex action. The latter function has the important property that it contains information about the lengths of all the geodesics joining any two points. In this case the expression of the heat kernel has an integral representation that in general cannot be reduced to a function. However, in the case of the Kolmogorov operator this integral can be computed explicitly, and the result obtained is a function-type heat kernel. Despite its robustness, the geometric method has its own difficulties. It is not always easy to find the action function, since this is a solution of a nonlinear equation, called the Hamilton–Jacobi equation. In the Riemannian case one can use other equivalent formalisms to find the action function, such as the Lagrangian formalism and the Hamiltonian formalism. In either case, one ends up solving a system of ODEs with boundary conditions. In the case of sub-Riemannian geometry the aforementioned formalisms are not always equivalent, since they produce possibly distinct geodesics (regular and normal geodesics, respectively). It is believed that the normal geodesics are the ones useful in computing the heat kernel. Even if the action function has a reasonable expression, the transport equation, whose solution is the volume function, might not be easily solved explicitly, unless it is of a very particular type. The geometric method can be successfully applied in a number of cases when the Riemannian metric has a familiar expression and the transport equation has separable variables. The role of the Fourier transform. This method is used whenever one wants to eliminate a variable. One of the situations where the Fourier method was proved efficient was in computing the heat kernels for sub-elliptic operators with one missing direction. In this case the partial Fourier transform transforms the sub-elliptic operator into an elliptic one, for which the heat kernel can be obtained by any other method. Then an application of the inverse partial Fourier transform on the heat kernel of the elliptic operator provides the heat kernel for the sub-elliptic operator. The method of eigenfunction expansion. This method can be applied in the case when the operator has a discrete spectrum. For instance, the Laplace–Beltrami operator on a compact Riemannian manifold has this property. The formula for the heat kernel provided by this method involves infinite series, unless some generating formulas exist. Unfortunately, these are rare, and their occurrence deserves to be cherished. The limited number of cases when generating formulas can reduce the heat kernel series to a familiar function includes the generating formulas of Mehler, Hille–Hardy, and Poisson. In general, the method of eigenfunction expansion does not provide a closed-form expression for the kernel. In the case of sub-elliptic operators this method was not as successful as the geometric method.

Conclusions

421

The method of path integrals. In spite of its obscurity, this is the most remarkable of the methods, since it provides an integral formula over all the paths joining the endpoints, reminding us of the way radiation propagates between two given states, making the relationship with quantum mechanics. This path integral can be computed explicitly in some particular cases, by either van Vleck’s or Feynman– Kac’s formula. The first one provides the heat kernel as a function-type formula, reminding us of the geometric method, where the volume function is the square root of the van Vleck determinant. However, this formula is not successfully applicable to sub-elliptic operators. Explicitly working out heat kernels for sub-elliptic operators by path integrals, even for simple cases like the Grushin or Heisenberg operators, has yet to be accomplished at the moment. The stochastic method. Each of the differential operators considered is a generator for an associated Ito diffusion processes. The success of this method is based on the ability to compute the probability density function of the associated Ito diffusion. This method can be applied for both elliptic and sub-elliptic operators. The main difficulty of the method is solving the associated stochastic differential equation. This can be done easily for simple operators such as the Laplace, Grushin or Kolmogorov. In the last two of these cases the probability density is obtained by evaluating an expectation integral using Feynman–Kac’s formula. The stochastic methods have proved useful not only in the case of parabolic operators but also in solving Dirichlet problems. The method of Laguerre calculus. This is the symbolic tensor calculus induced by the Laguerre functions on the Heisenberg group. These functions have been used in the study of the twisted convolution, or equivalently, the Heisenberg convolution. The Laguerre calculus plays a role similar to the Fourier series for a reasonable function defined on the unit circle. In order to invert a sub-elliptic operator (not necessarily second-order) on the Heisenberg group, we need to invert the Laguerre tensor of the operator. The method was successful in computing the heat kernels for the Kohn Laplacian b D @Nb @Nb C @Nb @Nb and Paneitz operator. The advantage of this method over the others is its applicability to finding heat kernels to powers of sub-Laplacians, wave operators and higher-order operators as long as the Laguerre tensor is diagonalizable. However, the method is somewhat limited since it is heavily based on the group structure and orthogonality of the Laguerre functions. The method of pseudo-differential operators. Using the symbolic calculus of pseudo-differential operators, we retrieved the heat kernels of several operators, such as the Hermite, Grushin, Kolmogorov and step-2 sub-Laplacian N C i ˛T operators. The disadvantage of this method is its lim 12 .Z ZN C ZZ/ itation to strongly elliptic operators and some sub-elliptic operators defined on two-step Heisenberg manifolds. The technique cannot be applied to sub-Laplacians of step higher than 2. The sum-of-squares operators. One of the special classes of operators treated in this book is the sum of squares of n linear independent vector fields. If n is the dimension of the space, then the operator is elliptic. Otherwise, it is sub-elliptic. In the latter

422

Conclusions

case, if the vector fields and their iterated brackets generate the tangent space at each point, then we say the bracket generating property holds. If this holds, two things happen: the sum-of-squares operator is hypoelliptic (H¨ormander’s theorem), and the distribution generated by the vector fields has the global connectivity property (Chow’s theorem). If exactly one bracket is needed to generate the tangent space, then the operator is said to be of step 2. The prototype operator in this case is the Heisenberg operator. If exactly k  1 brackets are needed to generate the tangent space, then the operator is said to be of step k. Step k D 1 operators correspond to the elliptic case. The methods and techniques discussed in the present book apply for steps k D 1; 2. Even if most of the methods work theoretically for any step, we did not encounter any explicit example where the computation produces exact solutions. There are a few examples of superior step operators treated in the literature by some adhoc methods, which do not belong to any of the methods treated in this book. One of the further developments is to create techniques to approach these types of operators.

List of Frequently Used Notations and Symbols

Rn j˛j

jxj

The n-dimensional Euclidean space The length of the muti-index ˛1 C    C ˛n , where ˛ D .˛1 ; : : : ; ˛n / Partial derivative with respect to xk @˛x11    @˛xnn P 1=2 n 2 if x D .x1 ; : : : ; xn / 2 Rn kD1 xk

L2 .I / Sn rx divX ijk .x/ ıx0  1 ;  2 ; 3 ;  4 Xt X.t/ E.Xt / Cov.Xt ; Xu / dX t F .x1 ; x2 I t1 ; t2 / P .AjB/ Wt W(t) pt .x0 ; x/ Gt .x/ Ik.x/  L x.t/; x.t/ P Scl S(x(t)) d.x0 ; x/ H(x, p) KO

Measurable and square integrable functions on I n-dimensional unit sphere in RnC1 The gradient in the x-variable The divergence of the vector field X The Christoffel symbols The Gamma function The Dirac distribution centered at x0 The theta-functions One-dimensional stochastic process n-dimensional stochastic process .Xt1 ; : : : ; Xtn / The expected value operator of the random variable Xt The covariance function The increment of the stochastic process Xt within time dt The joint distribution function of random variables X1 and X2 The conditional probability of A given B The one-dimensional Brownian motion The n-dimensional Brownian motion .Wt1 ; : : : ; Wtn / The probability density function Gaussian distribution of parameter t The Bessel function of first kind of order k Lagrangian function L W TM ! R The action associated with the classical Lagrangian The action evaluated along the path x.t/ The Riemannian distance between x0 and x Hamiltonian function H W T  M ! R The integral kernel of the integral operator K

@xk @˛x

423

424

Px;yIt d m./  H2nC1 h2nC1 H G X K(x, y;t) Fy i [X, Y] hx; yi Hn .x/ L.a/ n TM H C 1 .M / H Id E4 e4 @ G  F Wk.p/ P˛ b e.tI x; / MC .F / M.F / j ıi

List of Frequently Used Notations and Symbols

The space of continuous paths from x0 to x within time t The Weyl measure on the path space Px;yIt The Laplace operator 12 .@2x1 C    C @2xn / The (2n C 1)-dimensional Heisenberg group The Lie algebra of the Heisenberg group H2nC1 P 2 The Heisenberg Laplacian 2n X kD1 k on the Heisenberg group H2nC1 The Grushin operator P 2 The sub-elliptic Laplacian m kD1 Xk on a manifold of dimension n>m Heat kernel—the amount of heat transferred from x0 to x within time t > 0 The partial Fourier transform with respect to y p The imaginary number 1 The bracket X Y P YX , with X , Y vector fields or operators The inner product nkD1 xk yk if x D .x1 ; : : : ; xn /; y D .y1 ; : : : ; yn / 2 Rn The Hermite polynomial of degree n The Laguerre polynomial of degree n and parameter a The tangent bundle of the manifold M A nonintegrable distribution—a sub-bundle of TM The set of smooth functions on the manifold M The set quaternion numbers The identity map The Engel group The Lie algebra of the Engel group The boundary of the set  The noncommutative twisted product of G and F The Cauchy–Szeg¨o kernel The Paneitz operator The Kohn Laplacian The Weyl symbol The positive Laguerre matrix of F The Laguerre tensor of the operator F The Kronecker delta function

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Index

action, 24, 61 function, 231 integral, 13 adjoint operator, 166 area element, 180

degenerate operator, 407 diffusion coefficient, 183 distribution function, 177 drift, 173, 174, 183 Dynkin’s formula, 156

Bessel operator, 176 process, 176 bicharacteristic curve, 24 bicharacteristics system, 231 bilinear generating function, 103 boundary conditions, 24 bracket generating, 206 Brownian motion, 152 Brownian motion, 159, 176, 185, 188, 196 on circle, 181 with drift, 168

eigenfunction, 89, 99 eigenfunctions expansion, 403 eigenvalue, 89, 386 eigenvectors, 409 elastic force, 22 electric field, 23 Engel group, 205, 220 Euclidean Laplacian, 266 metric, 273 Euler’s constant, 317 Euler-Lagrange equations, 24 existence, 153, 194 expectation operator, 164 exponential map, 226 exponential martingale, 165

Campbell-Hausdorff formula, 207 Cauchy’s problem, 196, 372 Cauchy-Szeg¨o kernels, 295 characteristic set, 382, 387 Chow’s condition, 201 classical action, 24 Clifford algebra, 218 module, 218 compact support, 90 complete orthogonal system, 89, 99 complex Hamilton-Jacobi theory, 229 conditional expectation, 196 conditional distribution, 179 conservative system, 24 convolution operator, 289 CR-pluriharmonic functions, 310

Feynman-Kac formula, 195 fiber integration, 263 Fourier transform, 97, 307, 316, 355 fundamental solution, 407, 410, 415

Galileo’s equation, 22 Gaussian density, 168 Gaussian distribution, 175 Gaussian random variable, 163 generalized Brownian motion, 168 generalized transport equation, 237, 242

431

432 generating function, 92 generating function formula, 294 generator, 154, 170, 174, 184 of an Ito diffusion, 154 geometric Brownian motion, 171 gravitational field, 21 force, 22 potential, 97 Green functions, 206 Grushin operator, 263, 265, 267, 269, 335, 407 plane, 265 type operator, 263, 264

H¨ormander condition, 201, 238, 244, 263 Haar measure, 204, 209, 292 Hamilton-Jacobi equation, 24, 232, 246 Hamiltonian, 61 function, 23 mechanics, 23 Hamiltonian system, 232 harmonic oscillator, 220 polynomials, 275, 278 heat kernel, 403, 415 Heaviside function, 190 Heisenberg brush, 302 group, 87, 209, 256, 267, 289, 309, 333 Laplacian, 87 sub-Laplacian, 334 type algebra, 227 type group, 261 type Lie algebra, 216 vector fields, 87 Hermite function, 353 operator, 58, 333, 400, 403 polynomial, 91 Hessian matrix, 378 Hessian matrix, 382 Hille-Hardy’s formula, 94 homogeneous polynomial, 394 horizontal Laplacian, 277 hypoellipticity, 203, 238

integrated Brownian motion, 162 isotropic Heisenberg group, 328 Ito diffusion, 152, 167, 188, 196 Ito’s formula, 153, 155, 176, 181

Index Jacobi’s theta-function, 98 transformation, 98

K¨ahler form, 277 kinetic energy, 13 Kohn Laplacian, 291, 306, 349 Kolmogorov operator, 335, 342 Kolmogorov’s backward equation, 156, 196 forward equation, 166 operator, 188 Kolmogorov’s operator, 415 Kronecker delta function, 91, 297

Lagrangian, 13, 23 mechanics, 13 Laguerre calculus, 289, 307 functions, 296 matrix, 296 polynomials, 294, 300, 319 series expansion, 301 tensor forms, 298 Laguerre polynomial, 95 Langevin’s equation, 167 Laplace operator, 341 Laplacian, 99, 104, 196, 376 Legendre function, 104 polynomial, 102 Lewy operator, 291 Lie algebra, 208 Lie algebra, 290 lift, 24 linear noise, 170, 194 linear oscillator, 22 Lipschitz condition, 153 lognormal distributed, 159 distribution, 171 lognormal distributed, 172

mean, 168, 174 measurable functions, 152 measure space, 227 Mehler’s formula, 91, 231 modified Bessel equation, 178 modified Bessel function, 95 multi-index, 362

Index multi-product formula, 363 multiple solutions, 195

nilpotence class, 140 nilpotent Lie algebra, 214 Lie group, 219, 259, 410 Lie groups, 205, 207 nonholonomic sub-bundle, 204 normal distribution, 175

oscillatory integral, 363, 378

Paneitz operator, 292, 309, 320 parabolic equations, 397 partial Fourier transform, 228 path integrals, 230 pendulum string, 22 Poisson’s summation formula, 98 polar coordinates, 97, 180 Popp’s measure, 206 potential energy, 13 function, 196 power series, 93 principal bundle, 275 symbol, 232 principal value convolution, 293 probability density, 175 pseudo-differential operator, 362, 372

quadratic polynomial, 397 quaternion numbers, 273 quaternionic Heisenberg group, 213, 255

Riccati equation, 334, 339 Riemann-Stieltjes integral, 163 Riemannian manifold, 13 metric, 204, 211, 222 volume form, 274 Rodrigues’ formula, 94, 102

433 simple pendulum, 22 Sobolev space, 294 spectral decomposition, 225 spectrum, 227 spherical Grushin operator, 282, 285 spherical harmonics, 295 squared Bessel process, 192 state, 13 stochastic process, 152, 163 sub-elliptic estimation, 382 sub-Laplacian, 203, 232, 254, 282, 408, 410 sub-Riemannian manifold, 202, 263 metric, 204 structure, 201, 219, 263 submersion, 263 symbol, 363, 372, 395, 407 symbolic calculus, 382

tangent bundle, 13 Taylor expansion, 178, 379 Taylor’s formula, 92 thermodynamical systems, 202 theta-function, 7 trajectory, 13 transition density, 158, 171, 176, 184, 186, 194 transition distribution, 163 twisted convolution operator, 307, 318

uniqueness, 153, 194

Van Vleck determinant, 249 variance, 168, 174 vector bundle, 263 Volterra’s integral equation, 374 volume element, 204, 231, 242, 308 form, 263

Wald’s distribution, 177 Weyl symbol, 377, 381–383, 400, 401, 403, 411 Wiener integral, 161, 174

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