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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.
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