Beyond Partial Differential Equations: On Linear and Quasi-Linear Abstract Hyperbolic Evolution Equations (Lecture Notes in Mathematics, 1898) - Softcover

Beyer, Horst Reinhard

Softcover

ISBN 10: 3540711287 ISBN 13: 9783540711285

Publisher: Springer, 2007

Synopsis

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.

About this title

From the Back Cover

The present volume is self-contained and introduces to the treatment of linear and nonlinear (quasi-linear) abstract evolution equations by methods from the theory of strongly continuous semigroups. The theoretical part is accessible to graduate students with basic knowledge in functional analysis. Only some examples require more specialized knowledge from the spectral theory of linear, self-adjoint operators in Hilbert spaces. Particular stress is on equations of the hyperbolic type since considerably less often treated in the literature. Also, evolution equations from fundamental physics need to be compatible with the theory of special relativity and therefore are of hyperbolic type. Throughout, detailed applications are given to hyperbolic partial differential equations occurring in problems of current theoretical physics, in particular to Hermitian hyperbolic systems. This volume is thus also of interest to readers from theoretical physics.

Edition details

• Publisher: Springer
• Publication date: 2007
• Language: English
• ISBN 10: 3540711287
• ISBN 13: 9783540711285
• Binding: Paperback
• Number of pages: 297