It may be that the greatest practical benefit of semiclassical methods
is conceptual. Certainly semiclassical notions inform the way one
looks at the qualitative features of wave functions, the
eigenfunction-eigenvalue problem, and problems in numerical analysis
such as choice and rates of convergence of basis sets. In this talk I
will report on some earlier work of ours on semiclassical approaches
to systems of coupled wave equations, the recent beautiful application
of some of this work by Zhilinskii et al to the qualitative and
topological understanding of band structure in the rovibrational
spectra of molecules, and our more recent efforts in this area on
semiclassical approximations on the angular momentum sphere and other
curved phase spaces.
