Overview

This book focuses on unstable systems both from the classical and the quantum mechanical points of view and studies the relations between them. The first part deals with quantum systems. Here the main generally used methods today, such as the Gamow approach, and the Wigner-Weisskopf method, are critically discussed. The quantum  mechanical Lax-Phillips theory developed by the authors, based on the dilation theory of Nagy and Foias and its more general extension to approximate semigroup evolution is explained. The second part provides a description of approaches to classical stability analysis and introduces geometrical methods recently developed by the authors, which are shown to be highly effective in diagnosing instability and, in many cases, chaotic behavior. It is  then shown that, in the framework of  the theory of symplectic manifolds, there is a systematic algorithm for the construction of a canonical transformation of any standard potential model Hamiltonian to geometric form, making accessible powerful geometric methods for stability analysis in a wide range of applications.

Full Product Details

Author:   Lawrence Horwitz ,  Yosef Strauss
Publisher:   Springer Nature Switzerland AG
Imprint:   Springer Nature Switzerland AG
Edition:   1st ed. 2020
Weight:   0.518kg
ISBN:  

9783030315696

ISBN 10:   303031569
Pages:   221
Publication Date:   12 June 2020
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

Part I: Quantum Systems and Their Evolution.- Chapter 1: Gamow approach to the unstable quantum system. Wigner-Weisskopf formulation. Analyticity and the propagator. Approximate exponential decay. Rotation of Spectrum to define states. Difficulties in the case of two or more final states.- Chapter 2: Rigged Hilbert spaces (Gel'fand Triples). Work of Bohm and Gadella. Work of Sigal and Horwitz, Baumgartel. Advantages and problems of the method.- Chapter 3: Ideas of Nagy and Foias, invariant subspaces. Lax-Phillips Theory (exact semigroup). Generalization to quantum theory (unbounded spectrum). Stark effect.- Relativistic Lee-Friedrichs model.- Generalization to positive spectrum.- Relation to Brownian motion, wave function collapse.- Resonances of particles and fields with spin. Resonances of nonabelian gauge fields.- Resonances of the matter fields giving rise to the gauge fields. Resonence of the two dimensional lattice of graphene. Part II: Classical Systems.- Chapter 4: General dynamical systems and instability. Hamiltonian dynamical systems and instability. Geometrical ermbedding of Hamiltonian dynamical systems. Criterion for instability and chaos, geodesic deviation.  Part III: Quantization.- Chapter 5: Second Quantization of geometric deviation. Dynamical instability. Dilation along a geodesic.- Part IV: Applications.- Chapter 6: Phonons. Resonances in semiconductors. Superconductivity (Cooper pairs). Properties of grapheme. Thermodynamic properties of chaotic  systems. Gravitational waves.

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