Quantum chaos on a toral phase space for general boundary conditions: recent new results

This paper is a brief review of recent new results for the quantum-chaos pr oblem on a 2D toral phase space for general boundary conditions (BCs) on th e torus. The following results will be reviewed and discussed: (a) The new concept of bund distribution for Hamiltonians periodic in phase space. (b) Expressions for the general smooth torus map and for the condition determin ing the allowed quantum BCs associated with this map. (c) An exact renormal ization scheme for quantum Anosov maps on the torus, which "eliminates" the general-BCs problem, replacing it by a fixed-BCs one. (C) 2001 Elsevier Sc ience B.V. All rights reserved.