Quantum scars on a sphere

Specific features of a quantum map that can be connected to periodic orbits of the corresponding classical system are investigated. The model we studied is the kicked top for which it is known that many statistical properties of the quantum system, such as distributions of eigenvalues and eigenvectors, are well described by the random matrix theory. It is shown that other statistical measures of eigenvector statistics (e.g., Shannon entropy) reveal deviations from predictions based on ensembles of Gaussian matrices. These differences may be associated with periodic orbits. A formula, valid for an arbitrary quantum map, which can be helpful in associating scarred wave functions with a particular periodic orbit, is given. A comparison is made with the semiclassical results.