Bernard Helffer
CAMBRIDGE UNIVERSITY PRESS 2013, 260 PAGES
PRICE (HARDBACK) £40.00 ISBN 978-1-107-03230-9
The aim of the book is to introduce various aspects of spectral analysis and to apply the theory to examples from different branches of physics, including Schrödinger operators and statistical physics. The book contains sixteen chapters, covering various aspects of the theory and an introduction. Every chapter, except the last, describes some aspect of spectral theory and includes a number of problems at the end to test the reader’s understanding. The final chapter contains many problems intended to challenge the reader’s understanding of the previous material.
A description of the chapters follows.
Chapter 2 covers unbounded operators, adjoints and self-adjoint operators. The correct approach to restricting the operators domain is considered from the viewpoints of the right notion of continuity and the maximal domain for the restricted operator.
Chapter 3 covers representation theorems. Identifying the space of continuous linear forms on H is done via the Riesz theorem. Two Hilbertian scalar products are also considered, leading to some non-trivial results.
Chapter 4 considers semibounded operators and the Friedrichs extension with respect to symmetric operators.
Chapter 5 looks at some basic general properties of compact operators with particular focus on examples.
Chapter 6 introduces spectral theory for linear operators in L(H) in the infinite dimensional case.
Chapter 7 presents a deeper analysis of some previous examples; one such example is from statistical mechanics, the rest demonstrate methods of dealing with unbounded operators but the inverse operator enables compact self-adjoint operator spectral theory to be used.
Chapter 8 is an extension of Chapter 7; here a replacement to the diagonalisation theorem is presented, this is the spectral theorem.
Chapter 9 propounds on the subject of essentially self adjoint operators and displays assumptions that mostly ensure the self adjoint operator associated with a partial differential equation is unique.
Chapter 10 considers the discrete and essential spectra of a self adjoint operator.
Chapter 11 describes the max-min principle, used to describe the lowest part of a discrete spectrum.
Chapter 12 applies spectral methods to the Rayleigh equation of fluid mechanics.
Chapter 13 introduces pseudospectra for use with non-self-adjoint operators. The problem being that the spectrum is too unstable with respect to perturbations.
Chapter 14 applies the earlier theory to non-self-adjoint one dimensional models, presenting different approaches to enable generalisation to higher dimensional models.
Chapter 15 contains an analysis of the main spectral properties of the Kramers-Fokker-Planck operator of kinetic theory.
Chapter 16 provides more challenging problems to test the reader’s understanding of the book’s material.
This book is aimed at graduates who have already gained some knowledge of Hilbert spaces; I would recommend the book for such students.
John Bartlett CMath MIMA
Book review published directly onto IMA website (October 2014)
