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**Mathematical analysis and numerical methods for sciences and technology. (In six volumes). Volume 3: Spectral theory and applications. With the collaboration of Michel Artola and Michel Cessenat. Transl. from the French by John C. Amson.**
*(English)*
Zbl 0766.47001

Berlin etc.: Springer-Verlag. x, 515 p. (1990).

This nicely edited volume as already the remaining ones in the series of six dealing with various fundamental aspects of modern mathematical tools for physical and engineering sciences, consists of two comprehensive chapters. To each of them is appended one appendix. The first chapter of volume 3 is purely mathematical and deals with fundamental aspects of the spectral theory of linear operators in Banach and particularly Hilbert spaces. Firstly, such fundamental notions as the resolvent operator and the spectrum are introduced and their general properties are investigated. Next, the spectral decomposition of self-adjoint and compact normal operators is studied in detail, including a self-adjoint operator with compact inverse and min-max principle. This part is illustrated by many elaborated examples related to Sturm-Liouville problems and the Laplacian. For instance, Legendre, Laguerre, Hermite and Chebyshev polynomials are introduced in a natural fashion. Another class of problems is related to the decomposition of bounded or unbounded self- adjoint operators in a complex separable Hilbert space. That part of the book contains one of the most important results presented in an accessible way. Having examined the spectral decomposition of a self- adjoint operator, one can pass to the study of functions of such an operator. For a strictly positive self-adjoint operator the notion of fractional powers is applied to the construction of spaces intermediate between Hilbert spaces.

The Hilbert integral, being a Hilbert space, is introduced and its main properties are studied. Its usefulness for evolution, or initial value problems is exhibited.

Simple examples of unbounded self-adjoint operators and of unitary operators with a continuous spectrum reveal that no eigenvector in the primal Hilbert space may exist. Hence the need for generalized eigenvectors, for instance distributions. The relevant notions are presented in a precise fashion.

The appendix to the first chapter of the book includes the strong and weak Krein-Rutman theorems which deal with the specific properties of eigenvalues and eigenvectors for linear and compact positive operators. To illustrate them, these theorems are applied to the Laplacian in the Dirichlet problem and the diffusion multigroup problem in the neutron theory.

The comprehensive second chapter deals with applications of the general results presented previously to selected problems of theoretical and mathematical physics such as the static electromagnetism and the quantum physics. Those applications are preceded by a unified and consequent presentation of the properties of the gradient, divergence and curl operators and of the relevant spaces, being subspaces of \(L^ 2\). The reviewer finds this introductory yet rather deeply elaborated part very useful since it gathers the results otherwise scattered in the literature. It can be strongly recommended also to researchers involved in the study of mathematical aspects of the mechanics of continuous media.

The spectral theory developed in the first chapter of the book is not sufficient for the study of operators arising in the quantum physics. The indispensable complementary notions such as \(C^*\)-algebras, spectral measures and von Neumann algebras are presented in a rather abstract way in a separate appendix.

The english terminology used throughout the book is not always the commonly accepted one. For instance the word “denseness” is used instead of the “density”; sometimes the word “frontier” replaces the correct one, namely the “boundary”. The list of references is far from being exhaustive what is somewhat surprising since such a solid series should be equipped with a solid list of references. Even the famous book of S. Banach on linear operators has not been included.

The book is addressed mainly to applied mathematicians and theoretical physicists. Additionally, a certain part presents great interest to mathematically oriented specialists in the mechanics of continuous media. The interested reader will certainly enjoy studying the book, though to profit he must be prepared for a hard work. [The French original of this book and the other volumes of this series have been reviewed in Zbl 0642.35001, Zbl 0664.47002, Zbl 0664.47003 and Zbl 0652.45001.].

The Hilbert integral, being a Hilbert space, is introduced and its main properties are studied. Its usefulness for evolution, or initial value problems is exhibited.

Simple examples of unbounded self-adjoint operators and of unitary operators with a continuous spectrum reveal that no eigenvector in the primal Hilbert space may exist. Hence the need for generalized eigenvectors, for instance distributions. The relevant notions are presented in a precise fashion.

The appendix to the first chapter of the book includes the strong and weak Krein-Rutman theorems which deal with the specific properties of eigenvalues and eigenvectors for linear and compact positive operators. To illustrate them, these theorems are applied to the Laplacian in the Dirichlet problem and the diffusion multigroup problem in the neutron theory.

The comprehensive second chapter deals with applications of the general results presented previously to selected problems of theoretical and mathematical physics such as the static electromagnetism and the quantum physics. Those applications are preceded by a unified and consequent presentation of the properties of the gradient, divergence and curl operators and of the relevant spaces, being subspaces of \(L^ 2\). The reviewer finds this introductory yet rather deeply elaborated part very useful since it gathers the results otherwise scattered in the literature. It can be strongly recommended also to researchers involved in the study of mathematical aspects of the mechanics of continuous media.

The spectral theory developed in the first chapter of the book is not sufficient for the study of operators arising in the quantum physics. The indispensable complementary notions such as \(C^*\)-algebras, spectral measures and von Neumann algebras are presented in a rather abstract way in a separate appendix.

The english terminology used throughout the book is not always the commonly accepted one. For instance the word “denseness” is used instead of the “density”; sometimes the word “frontier” replaces the correct one, namely the “boundary”. The list of references is far from being exhaustive what is somewhat surprising since such a solid series should be equipped with a solid list of references. Even the famous book of S. Banach on linear operators has not been included.

The book is addressed mainly to applied mathematicians and theoretical physicists. Additionally, a certain part presents great interest to mathematically oriented specialists in the mechanics of continuous media. The interested reader will certainly enjoy studying the book, though to profit he must be prepared for a hard work. [The French original of this book and the other volumes of this series have been reviewed in Zbl 0642.35001, Zbl 0664.47002, Zbl 0664.47003 and Zbl 0652.45001.].

Reviewer: J.J.Telega (Warszawa)

### MSC:

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

47A70 | (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces |

47B25 | Linear symmetric and selfadjoint operators (unbounded) |

47A10 | Spectrum, resolvent |

47B40 | Spectral operators, decomposable operators, well-bounded operators, etc. |

78A30 | Electro- and magnetostatics |

46L05 | General theory of \(C^*\)-algebras |

81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |

46L10 | General theory of von Neumann algebras |

### Keywords:

Hamiltonian operators in quantum physics; \(C^*\)-algebras; spectral theory of linear operators; resolvent operator; spectrum; spectral decomposition of self-adjoint and compact normal operators; self-adjoint operator with compact inverse; min-max principle; Sturm-Liouville problems; Laplacian; Legendre, Laguerre, Hermite and Chebyshev polynomials; fractional powers; spaces intermediate between Hilbert spaces; Hilbert integral; generalized eigenvectors; Krein-Rutman theorems; compact positive operators; Dirichlet problem; diffusion multigroup problem in the neutron theory; static electromagnetism; gradient, divergence and curl operators; mechanics of continuous media; spectral measures; von Neumann algebras
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\textit{R. Dautray} and \textit{J.-L. Lions}, Mathematical analysis and numerical methods for sciences and technology. (In six volumes). Volume 3: Spectral theory and applications. With the collaboration of Michel Artola and Michel Cessenat. Transl. from the French by John C. Amson. Berlin etc.: Springer-Verlag (1990; Zbl 0766.47001)