Semigroup Theory uses abstract methods of Operator Theory to treat initial bou- ary value problems for linear and nonlinear equations that describe the evolution of a system. Due to the generality of its methods, the class of systems that can be treated in this way exceeds by far those described by equations containing only local op- ators induced by partial derivatives, i.e., PDEs. In particular, that class includes the systems of Quantum Theory. Another important application of semigroup methods is in ?eld quantization. Simple examples are given by the cases of free ?elds in Minkowski spacetime like Klein-Gordon ?elds, the Dirac ?eld and the Maxwell ?eld, whose ?eld equations are given by systems of linear PDEs. The second quantization of such a ?eld replaces the ?eld equation by a Schrodinger ¨ equation whose Hamilton operator is given by the second quantization of a non-local function of a self-adjoint linear operator. That operator generates the semigroup given by the time-development of the solutions of the ?eld equation corresponding to arbitrary initial data as a function of time.
Rezensionen / Stimmen
From the reviews:
"In the monograph under review, designed as a textbook for graduate students, the author aims to build up the theory in a straightforward way in view of physical applications. . The material covered in the book was used in a two-semester course for graduate students. The book is a good summary of the most important facts of the theory of evolution equations, and is also a good source of information for researchers looking for new applications of this theory." (András Bátkai, Mathematical Reviews, Issue 2008 h)
"This well-written work is the outgrowth of a two-semester course on linear semigroup methods, linear and quasi-linear hyperbolic systems of partial differential equations, and the abstract theory of evolution equations . . Although the text was written for truly advanced graduate students, it contains a wealth of well presented results on semigroup theory of evolution equations . and will therefore serve as a valuable resource for researchers in mathematics and theoretical physics as well." (Thomas Hagen, Zentralblatt MATH, Vol. 1144, 2008)
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ISBN-13
978-3-540-71129-2 (9783540711292)
DOI
10.1007/978-3-540-71129-2
Schweitzer Klassifikation
Conventions.- Mathematical Introduction.- Prerequisites.- Strongly Continuous Semigroups.- Examples of Generators of Strongly Continuous Semigroups.- Intertwining Relations, Operator Homomorphisms.- Examples of Constrained Systems.- Kernels, Chains, and Evolution Operators.- The Linear Evolution Equation.- Examples of Linear Evolution Equations.- The Quasi-Linear Evolution Equation.- Examples of Quasi-Linear Evolution Equations.
