  We discuss several problems in quasiclassical physics for which approximate
solutions were recently obtained by a new method, and which can also be solved
by novel versions of the Born-Oppenheimer approximation. These cases include
the so-called bouncing ball modes, low angular momentum states in perturbed
circular billiards, resonant states in perturbed rectangular billiards, and
whispering gallery modes. Some rare, special eigenstates, concentrated close to
the edge or along a diagonal of a nearly rectangular billiard are found. This
kind of state has apparently previously escaped notice.
