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Synopsis
This is the first monograph to present a comprehensive treatment of the mathematical foundations of quantum chaos. Precise results in this area involve an exciting mixture of analytical number theory, zeta and L-functions, random matrix theory, scattering theory, the Selberg trace formula, and related global functional analysis. Many examples are presented including polygonal and standard billiards systems and models on the pseudosphere. The physics of both compact and finite volume systems are discussed, as well as systems in the presence of a magnetic field. Results on the spectra of Gutzwiller models for mesoscopic systems are discussed including questions of dissolving eigenvalues, simplicity of the spectra and exceptional eigenvalues. Relationships to isometric-isospectral questions in physics are discussed. Finally, applications of quantum chaos to recent results on mesoscopic physics are discussed, in particular transport properties in these devices.
Starting from simple examples, the text leads the reader through the most recent work of Sarnak, Luo and coworkers on arithmetic chaos, Zelditch, Degli Esposti and coworkers on quantum ergodicity, Bleher and coworkers on integrable systems, Gutkin, Veech and coworkers on polygonal billiards, Sarnak, Phillips and coworkers on spectra of Gutzwiller models, Mueller and others on scattering theory, Berry, Keating, Steiner, Aurich, Bolte, Schmit, Bogomolny and coworkers on quantum chaos and Marcus Beenakker and coworkers on mesoscopic systems.
Audience: This book will be of use to physicists, mathematicians, and engineers interested in quantum chaos and its applications to mesoscopic systems.
Booknews
The author poses the question: "...What are the signatures of quantum chaos?" In answer, he provides an advanced level tour of fingerprints of classical chaos systems in the quantum spectra via text examples, theorems, and equations. A sample of covered topics from mathematical physics and related fields includes random matrix theory, analytical number theory, scattering theory, and quantum ergodic systems. An especially intriguing experimental testing ground for chaos theory is the use of microwave billiards to study the transition from integrable to chaotic systems and its theoretical and engineering applications to mesoscopic systems. Annotation c. by Book News, Inc., Portland, Or.