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 methods are critically described, such as the Gamow approach, the Wigner-Weisskopf formulation, the Lax-Phillips theory, and a method developed by the authors using the dilation construction proposed by Nagy and Foias. The second part provides a description of approaches to classical stability analysis and introduces geometrical methods recently developed by the authors, which show to be highly effective in diagnosing instability and, in many cases, chaotic behavior. Part three shows that many of the aspects of the classical picture display properties that can be associated with underlying quantum phenomena, as should be expected in the real world.

Contenu

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.