##
**Introduction to spectral theory. With applications to Schrödinger operators.**
*(English)*
Zbl 0855.47002

Applied Mathematical Sciences. 113. New York, NY: Springer-Verlag. xi, 501 p. (1996).

The book emphasizes the geometric aspect of spectral analysis where spectral properties of operators are investigated by studying these operators on families of functions having certain geometric support conditions. It presents a modern overview of this geometric spectral analysis.

The theory of linear operators in Hilbert spaces is introduced in some detail in the beginning of this book. This part is standard and furnishes the necessary mathematical background to tackle the remainder of the book; it may be used as a guideline by the interested readership. The book concerns itself mainly with the discrete and essential parts of the spectrum, although embedded eigenvalues are also introduced. This general operator-theoretical part is illustrated by examples taken from the theory of Schrödinger operators, such as the exponential decay of eigenfunctions in terms of the Agmon metric, or to prove the essential selfadjointness, local compactness, or relative boundedness for certain operators. This part also includes standard spectral stability results for the discrete and essential spectrum. However, several results are also mentioned which are not so well-known in the textbooks. One example is Perssons theorem which gives a formula for the bottom of the essential spectrum. Moreover, parts of semiclassical analysis are given, for instance the semiclassical limit of eigenvalues is studied as well as quantum tunneling and double-well potentials. The book does not contain semiclassical analysis in the context of microlocal analysis developed e.g. by Helffer, Maslov, or Robert.

The main part of the book (more than one third) consists of a collection of results in resonance theory. In the last two decades the theory of resonances has developed in several different directions. There exists a bulk of material and results in the research literature. The present book gives a worthwhile overview of these results. The main topics are spectral deformation, spectral stability, and nontrapping estimates. The theory of Aguilar, Balslev, Combes, Simon is explained in detail and applied to shape resonances. Spectral deformation theory is explained in \(\mathbb{R}^d\) and then applied to Schrödinger operators. Also a general theory for spectral stability is given. This is related to nonanalytic perturbation theory for discrete eigenvalues and to perturbations of embedded eigenvalues and resonances. Finally, some further topics and features from the recent resonance literature are given. The position of the resonance is related to the resonance width. Resonance phenomena arising also in the presence of an electric or magnetic field are described. Further topics in the quantum theory of resonances are mentioned.

The main feature and probably also the main objective of this book is the overview of a large part of resonance theory. It does not contain the approach of Helffer and Sjöstrand. However, it emphasizes the geometric spectral analytic aspect in this theory. It collects together important and recent results on resonances obtained during the last two decades. A mathematical introduction to the spectral and perturbation theory of selfadjoint operators makes this book selfcontained. Exercises are given which help to give a better understanding of the text. The appendices are in the main devoted to explaining the theory of linear operators in Banach spaces. Unfortunately, there is no list of symbols used in the text. The book is well written. It gives a selfconsistent guideline for further studies in resonance theory and geometric aspects of spectral theory. It will be useful for graduate students as well as for mathematicians and physicists interested in spectral theory. The book fills a gap in the present literature.

The theory of linear operators in Hilbert spaces is introduced in some detail in the beginning of this book. This part is standard and furnishes the necessary mathematical background to tackle the remainder of the book; it may be used as a guideline by the interested readership. The book concerns itself mainly with the discrete and essential parts of the spectrum, although embedded eigenvalues are also introduced. This general operator-theoretical part is illustrated by examples taken from the theory of Schrödinger operators, such as the exponential decay of eigenfunctions in terms of the Agmon metric, or to prove the essential selfadjointness, local compactness, or relative boundedness for certain operators. This part also includes standard spectral stability results for the discrete and essential spectrum. However, several results are also mentioned which are not so well-known in the textbooks. One example is Perssons theorem which gives a formula for the bottom of the essential spectrum. Moreover, parts of semiclassical analysis are given, for instance the semiclassical limit of eigenvalues is studied as well as quantum tunneling and double-well potentials. The book does not contain semiclassical analysis in the context of microlocal analysis developed e.g. by Helffer, Maslov, or Robert.

The main part of the book (more than one third) consists of a collection of results in resonance theory. In the last two decades the theory of resonances has developed in several different directions. There exists a bulk of material and results in the research literature. The present book gives a worthwhile overview of these results. The main topics are spectral deformation, spectral stability, and nontrapping estimates. The theory of Aguilar, Balslev, Combes, Simon is explained in detail and applied to shape resonances. Spectral deformation theory is explained in \(\mathbb{R}^d\) and then applied to Schrödinger operators. Also a general theory for spectral stability is given. This is related to nonanalytic perturbation theory for discrete eigenvalues and to perturbations of embedded eigenvalues and resonances. Finally, some further topics and features from the recent resonance literature are given. The position of the resonance is related to the resonance width. Resonance phenomena arising also in the presence of an electric or magnetic field are described. Further topics in the quantum theory of resonances are mentioned.

The main feature and probably also the main objective of this book is the overview of a large part of resonance theory. It does not contain the approach of Helffer and Sjöstrand. However, it emphasizes the geometric spectral analytic aspect in this theory. It collects together important and recent results on resonances obtained during the last two decades. A mathematical introduction to the spectral and perturbation theory of selfadjoint operators makes this book selfcontained. Exercises are given which help to give a better understanding of the text. The appendices are in the main devoted to explaining the theory of linear operators in Banach spaces. Unfortunately, there is no list of symbols used in the text. The book is well written. It gives a selfconsistent guideline for further studies in resonance theory and geometric aspects of spectral theory. It will be useful for graduate students as well as for mathematicians and physicists interested in spectral theory. The book fills a gap in the present literature.

Reviewer: M.Demuth (Clausthal)

### MSC:

47A10 | Spectrum, resolvent |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35Pxx | Spectral theory and eigenvalue problems for partial differential equations |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

35J10 | Schrödinger operator, Schrödinger equation |