Billiards are studied whose boundaries comprise two or more segments w
hich allow local (but not global) separation of the Helmholtz equation
. Related local solutions are labeled ''sep'' functions. Such billiard
s comprise the Sigma and Sigma sets; A definition of quantum chaos for
these sets of billiards is presented based on the ratio of the ''fluc
tuation length'' of the wave function nodal pattern to the ''c-diamete
r'' of the billiard. The ''function-mixing hypothesis'' states that a
sufficient condition for a billiard to be chaotic is that the billiard
be an element of one of these sets. It further ascribes such chaotic
behavior to be due to mixing of dissimilar sep functions. Examples of
the application of this hypothesis are described. A set-theoretic form
alism is introduced to describe perturbation theory for infinite poten
tials and applied to the concave-sided square billiard of sufficiently
small concavity. It is concluded that the adiabatic theorem of quantu
m mechanics does not apply to this configuration in the limit of large
quantum numbers.