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INTRODUCTION

very recent research to classify the symbols generating commutative C* algebras of

Toeplitz operators.

In the other hand, Yuri Egorov describes the theory of Pseudo-Differential

Operators, first in

JR.~

and then on manifolds. To do this he devotes two sections to

briefly explain the main ideas of classical and quantum mechanics involved in the

understanding of Pseudo-Differential Operators, making emphasis on the theory

of canonical transformations. In his last section, among other things, he explains

the use of Fourier Integral Operators when dealing with Differential and Pseudo-

Differential Operators.

Stephan De Bievre does a very good job explaining in rigorous mathematical

terms what it is currently understood by quantum chaos. This topic is a very active

research area in both Mathematics and Physics. The reader who is a mathematician

interested in learning the physics needed and involved in the quantum chaos theory

will highly appreciate the content of De Bievre' s lecture notes. He includes several

sections to explain the main ideas in classical, quantum mechanics, and the process

of quantization in a clear and rigorous mathematical manner. De Bievre motivates

the reader starting with the study of the classical flow on a billiard on a domain

0 in

JR.-"'

and the spectral problem of the Dirichlet Laplacian on 0. He shows how

important can be the regular or irregular properties of the classical flow to study

the spectral statistics of the spectrum of eigenvalues of the Dirichlet Laplacian. He

explains the Schnirelman theorem and the Berry-Tabor, Bohigas-Giannoni-Schmit

conjectures in this context. Then he introduces the concept of mixing as a measure

of irregularity for a Hamiltonian System on a given phase space (he shows how

billiards flows are Hamiltonian). In order to consider a workable model to illustrate

all these ideas, De Bievre considers the Torus regarded as the phase space. Then,

by introducing iterations of automorphisms on the torus (the discrete version of

Hamiltonian flows), he illustrates the existence of mixing and finally he does the

quantum mechanics on the Torus to illustrate and prove the Schnirelman theorem

in this context.

In chapter six, Pablo Padilla shows how variational methods can be used to

find solutions of non-linear problems by considering a suitable functional and the

corresponding Euler-Lagrange (EL) equations. First he obtains the (EL) equations

when the functional is determined by a Lagrangian and gives examples and exercises

in applications like the brachystochrone problem, the wave equation, equilibrium

and reaction-diffusion process, geodesics on manifolds and minimal surfaces. Then

by introducing the concept of second variation and null Lagrangians he proves the

Brower's fixed point theorem. Padilla also deals with the problem of existence

of minimizers and gives two very interesting applications: minimal hypersurfaces

in Riemannian manifolds and non-linear eigenvalue problems with constrains. He

finishes by considering the problem of finding solutions (other than minima) of a

non-linear partial differential equation as critical points of a related functional. To

do that he explains the minimax method, the Palais-Smale condition, and applies

these ideas to prove the Mountain-Pass theorem. The reader will find a nice mo-

tivation in Padilla's lecture notes to see how several branches of mathematics get

mixed in non-linear problems such as Topology, Geometry and Analysis.