The second edition of this textbook presents the basic mathematical
knowledge and skills that are needed for courses on modern theoretical
physics, such as those on quantum mechanics, classical and quantum field
theory, and related areas. The authors stress that learning mathematical
physics is not a passive process and include numerous detailed proofs,
examples, and over 200 exercises, as well as hints linking mathematical
concepts and results to the relevant physical concepts and theories. All
of the material from the first edition has been updated, and five new
chapters have been added on such topics as distributions, Hilbert space
operators, and variational methods.
The text is divided into three parts:
- Part I: A brief introduction to (Schwartz) distribution theory.
Elements from the theories of ultra distributions and (Fourier)
hyperfunctions are given in addition to some deeper results for Schwartz
distributions, thus providing a rather comprehensive introduction to the
theory of generalized functions. Basic properties and methods for
distributions are developed with applications to constant coefficient
ODEs and PDEs. The relation between distributions and holomorphic
functions is considered, as well as basic properties of Sobolev spaces.
- Part II: Fundamental facts about Hilbert spaces. The basic theory of
linear (bounded and unbounded) operators in Hilbert spaces and special
classes of linear operators - compact, Hilbert-Schmidt, trace class, and
Schrödinger operators, as needed in quantum physics and quantum
information theory - are explored. This section also contains a detailed
spectral analysis of all major classes of linear operators, including
completeness of generalized eigenfunctions, as well as of (completely)
positive mappings, in particular quantum operations.
- Part III: Direct methods of the calculus of variations and their
applications to boundary- and eigenvalue-problems for linear and
nonlinear partial differential operators. The authors conclude with a
discussion of the Hohenberg-Kohn variational principle.
The appendices contain proofs of more general and deeper results,
including completions, basic facts about metrizable Hausdorff locally
convex topological vector spaces, Baire's fundamental results and their
main consequences, and bilinear functionals.
Mathematical Methods in Physics is aimed at a broad community of
graduate students in mathematics, mathematical physics, quantum
information theory, physics and engineering, as well as researchers in
these disciplines. Expanded content and relevant updates will make this
new edition a valuable resource for those working in these disciplines.
