  Band Husimi distributions (BHDs) are introduced in the quantum-chaos problem
on a toral phase space. In the framework of this phase space, a quantum state
must satisfy Bloch boundary conditions (BCs) on a torus and the spectrum
consists of a finite number of levels for given BCs. As the BCs are varied, a
level broadens into a band. The BHD for a band is defined as the uniform
average of the Husimi distributions for all the eigenstates in the band. The
generalized BHD for a set of adjacent bands is the average of the BHDs
associated with these bands. BHDs are shown to be closer, in several aspects,
to classical distributions than Husimi distributions for individual
eigenstates. The generalized BHD for two adjacent bands is shown to be
approximately conserved in the passage through a degeneracy between the bands
as a nonintegrability parameter is varied. Finally, it is shown how generalized
BHDs can be defined so as to achieve physical continuity under small variations
of the scaled Planck constant. A generalization of the topological
(Chern-index) characterization of the classical-quantum correspondence is then
obtained.
