Function-mixing hypothesis and quantum chaos

Abstract

Billiards are studied whose boundaries comprise two or more segments which allow local (but not global) separation of the Helmholtz equation. Related local solutions are labeled “sep” functions. Such billiards comprise the Σ and Σ sets. A definition of quantum chaos for these sets of billiards is presented based on the ratio of the “fluctuation length” of the wave function nodal pattern to the “c-diameter” 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 formalism is introduced to describe perturbation theory for infinite potentials and applied to the concave-sided square billiard of sufficiently small concavity. It is concluded that the adiabatic theorem of quantum mechanics does not apply to this configuration in the limit of large quantum numbers.