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 problem on a 2D toral phase space for general boundary conditions (BCs) on the torus. The following results will be reviewed and discussed: (a) The new concept of band distribution for Hamiltonians periodic in phase space. (b) Expressions for the general smooth torus map and for the condition determining the allowed quantum BCs associated with this map. (c) An exact renormalization scheme for quantum Anosov maps on the torus, which “eliminates” the general-BCs problem, replacing it by a fixed-BCs one.