Hyperbolic geometry is a classical subject in pure mathematics which has exciting applications in theoretical physics. In this book leading experts introduce hyperbolic geometry and Maass waveforms and discuss applications in quantum chaos and cosmology. The book begins with an introductory chapter detailing the geometry of hyperbolic surfaces and includes numerous worked examples and exercises to give the reader a solid foundation for the rest of the book. In later chapters the classical version of Selberg's trace formula is derived in detail and transfer operators are developed as tools in the spectral theory of Laplace-Beltrami operators on modular surfaces. The computation of Maass waveforms and associated eigenvalues of the hyperbolic Laplacian on hyperbolic manifolds are also presented in a comprehensive way. This book will be valuable to graduate students and young researchers, as well as for those experienced scientists who want a detailed exposition of the subject.
Jens Bolte joined the Department of Mathematics at Royal Holloway, University of London, in 2007. He works in the field of quantum chaos and is, in particular, interested in arithmetic quantum chaos, semiclassical quantum mechanics and quantum graph models. Frank Steiner is a Professor at Ulm University. He has spent sabbaticals at CERN and the Universities of Geneva, Lausanne, Paris and Princeton. His present areas of research are quantum graph models and cosmology.
Preface; 1. Hyperbolic geometry A. Aigon-Dupuy, P. Buser and K.-D. Semmler; 2. Selberg's trace formula: an introduction J. Marklof; 3. Semiclassical approach to spectral correlation functions M. Sieber; 4. Transfer operators, the Selberg Zeta function and the Lewis-Zagier theory of period functions D. H. Mayer; 5. On the calculation of Maass cusp forms D. A. Hejhal; 6. Maass waveforms on (? 0(N), x) (computational aspects) Fredrik Stroemberg; 7. Numerical computation of Maass waveforms and an application to cosmology R. Aurich, F. Steiner and H. Then.