Mathematical methods in physics. Distributions, Hilbert space operators, variational methods, and applications in quantum physics.
2nd ed.

*(English)*Zbl 1330.46001
Progress in Mathematical Physics 69. Cham: Birkhäuser/Springer (ISBN 978-3-319-14044-5/hbk; 978-3-319-14045-2/ebook). xxvii, 597 p. (2015).

This book gives a detailed survey on mathematical methods in physics and treats the following three important parts:

\(\bullet\) Distributions,

\(\bullet\) Hilbert space operators,

\(\bullet\) Variational methods.

The book is very suitable for students of physics, mathematics or engineering with a good background in analysis and linear algebra.

The first part starts with an introduction to the theory of Schwartz distributions. In the following chapters, among other things, tensor products, convolution products and Fourier transforms of distributions are defined and studied. There is one section on holomorphic functions and the theory of this class of functions. In this section, holomorphic functions are defined to be elements of the null space in \(C^\infty\) of the Cauchy-Riemann operator \[ \overline\partial:= {1\over 2}\Biggl({\partial\over\partial x}+ i{\partial\over\partial y}\Biggr). \] In a later section, distributions are characterized as finite sums of boundary values of holomorphic functions. The first part ends with a brief presentation of Sobolev spaces and their classical aspects according to later applications (e.g., the Dirichlet-Laplace operator in Section 36.2).

In the second part, Hilbert spaces and the well-known theory on this topic are introduced. It follows a very detailed treatment of linear operators (bounded and unbounded). In particular, compact, trace class and Hilbert-Schmidt operators are discussed. This part also includes a comprehensive spectral analysis of linear operators. Part 2 is completed by positive maps, their spectral theory and their importance in quantum physics.

Part 3 is devoted to variational methods. After a short introduction into this topic, minimization problems in Banach spaces are extensively discussed, especially the Lagrange multiplier method and its connection to eigenvalues of partial differential operators. Finally, the results are applied to the Hohenberg-Kohn density theory.

The appendices deal with the completion of metric spaces, the Baire category theorem and its applications like the uniform boundedness principle. The open mapping theorem, and the continuity of bilinear functionals on Hausdorff locally convex vector spaces are discussed.

Besides the main theory, almost each chapter of this book contains a variety of suitable exercises to challenge the readers. Also the historical remarks on the main topics should be emphasized.

All in all, the book has a high didactical and scientific quality so that it can be recommended for both graduate students and researchers.

For the first edition, see [ Zbl 1019.46001].

\(\bullet\) Distributions,

\(\bullet\) Hilbert space operators,

\(\bullet\) Variational methods.

The book is very suitable for students of physics, mathematics or engineering with a good background in analysis and linear algebra.

The first part starts with an introduction to the theory of Schwartz distributions. In the following chapters, among other things, tensor products, convolution products and Fourier transforms of distributions are defined and studied. There is one section on holomorphic functions and the theory of this class of functions. In this section, holomorphic functions are defined to be elements of the null space in \(C^\infty\) of the Cauchy-Riemann operator \[ \overline\partial:= {1\over 2}\Biggl({\partial\over\partial x}+ i{\partial\over\partial y}\Biggr). \] In a later section, distributions are characterized as finite sums of boundary values of holomorphic functions. The first part ends with a brief presentation of Sobolev spaces and their classical aspects according to later applications (e.g., the Dirichlet-Laplace operator in Section 36.2).

In the second part, Hilbert spaces and the well-known theory on this topic are introduced. It follows a very detailed treatment of linear operators (bounded and unbounded). In particular, compact, trace class and Hilbert-Schmidt operators are discussed. This part also includes a comprehensive spectral analysis of linear operators. Part 2 is completed by positive maps, their spectral theory and their importance in quantum physics.

Part 3 is devoted to variational methods. After a short introduction into this topic, minimization problems in Banach spaces are extensively discussed, especially the Lagrange multiplier method and its connection to eigenvalues of partial differential operators. Finally, the results are applied to the Hohenberg-Kohn density theory.

The appendices deal with the completion of metric spaces, the Baire category theorem and its applications like the uniform boundedness principle. The open mapping theorem, and the continuity of bilinear functionals on Hausdorff locally convex vector spaces are discussed.

Besides the main theory, almost each chapter of this book contains a variety of suitable exercises to challenge the readers. Also the historical remarks on the main topics should be emphasized.

All in all, the book has a high didactical and scientific quality so that it can be recommended for both graduate students and researchers.

For the first edition, see [ Zbl 1019.46001].

Reviewer: Michael Demuth (Goslar)

##### MSC:

46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |

46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |

46F05 | Topological linear spaces of test functions, distributions and ultradistributions |

47-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory |

47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |

47L05 | Linear spaces of operators |

47N50 | Applications of operator theory in the physical sciences |