Fringes decorating anticaustics in ergodic wavefunctions

The probability density Π is calculated for quantum eigenstates near spatial boundaries of classically chaotic regions. By contrast with integrable systems, for which the classical Π diverges on classical boundaries, which are caustics, in chaotic systems the classical Π does not diverge but vanishes abruptly in a way that depends on the number of freedoms N ; the boundaries are anticaustics. Quantum mechanics softens anticaustics, to give Π in terms of a set of canonical diffraction patterns, one for each N ; these are studied in detail. The appropriate definition of Π involves averaging over eigenstates in an energy range larger than O ( h ) but smaller than O ( h ⅔ ) (where h is Planck’s constant), that is over a range of ∆ N states near the N th, where N 1-1 / N ≪ ∆ N ≪ N 1-⅔ N .